text
stringlengths 6
976k
| token_count
float64 677
677
| cluster_id
int64 1
1
|
|---|---|---|
COMP 3009: Assignment 1 solved
Description
5/5 - (2 votes)
Figure 1: Illustration of the javelin hitting the shield
1 (50 points) Theory questions: Using vector operations and Transformations
1. (5 points) Given a point P = (Px, Py, Pz) and an object, O = (Ox, Oy, Oz),
1.1. Show the matrices used when computing the rotation O around P by angle α. The rotation is
around the X-axis.
1.2. What is the result when P = (1, 2, 3), O = (6, 5, 4) and α = 30o
. The rotation is around the
X-axis.
2. (3 points) Given two vectors v =< 4, 3, 2 > and u =< 5, 1, 7 >, find the angle between the two
vectors.
3. (8 points) What is the geometry matrix for this spline function? P(t) =
x(t)
y(t)
where x(t) = 2t
3 + 3t
2 − 6t + 1 and y(t) = 3t
3 − 4t
2 + 3t − 1.
4. (7 points) Given a triangle T p0 = (1, 1, 1); p1 = (−1, −1, −1); p2 = (−1, 1, −1);
1. Find the normal to the triangle. Assume that the triangle is given clockwise. Normalize the
vector. Draw a figure showing that your normal is correct.
2. Find the angle that is adjacent to p0.
3. Determine the area of the T
5. (8 points) A heroine is carrying a triangular shield as a protection from a javelin that was thrown
at her. The shield is designed to protect against any projectile that hit the shield at an angle of 60
degrees or less with respect to the shield surface, α ≤ 60 (α is the angle between the shield and the
javelin). The javelin velocity (towards the target) is v =< 0, −12, 11 > and it has hit the shield at
p = (px, py, pz). The shields coordinates are s0 = (5, 1, 0), s1 = (5, 2, 5) and s2 = (10, 3, 10). Figure 1
illustrates the question.
Determine whether the shield will protect the heroine. Show your work.
6. (7 points) Find the transformation matrix which is required to rotate an object around the vector
(p1 − p0) where p0 = (1, 1, 1) and p1 = (2, 3, 4). Show your work.
7. (6 points) Write an expression for the matrix that transforms the box drawn on the left to the box
on the right.
2 COMP 3009
8. (6 points) The matrix that converts a perspective view frustum to a canonical viewing volume is
provided below. What is the geometric interpretation for each parameter (draw a picture)?
Pre-processing
• Download the assignment code
• Compile and link the code – you should see a rotating sphere, a rotating cylinder, and a rotating
cube.
• Familiarize yourself with the code. Review how a cylinder and a sphere are created and used.
2 (10 points) Part 2 – Render a sphere
Purpose:
• View a simple 3D object.
• Experiment with model space transformation
• Familiarization with provided code
To do:
Create a project and modify code.
1. Modify the sphere to be at position 100, 10, 100.
2. Set up a view position (e.g., 200,200,200) and the look at vector to 100, 10, 100 (centre of sphere).
3 COMP 3009
3. The sphere is rotating around the y-axis. The rate of rotation is 0.1 degree per time step. Note that
the sphere is contracted along the y-axis.
4. Set up the sphere so that it rotates similarly to the earth (66.5 degrees). Determine which axis you
need to rotate around.
5. Display the sphere.
3 (40 points)Part 3 – Create a simple complex object (hierarchical
object)
Purpose:
• Create hierarchical object
• Use keyboard commands
• Use transformations
To do:
Create a simple hierarchical object and place it beside the sphere from Task 1 (Figure 2).
1. The object consists of a box and two spheres, where the spheres are placed on top of the box (Figure
2). Randomly set the three primary colours to the vertices (so that you can see the spheres rotating).
Use the cylinder code to create the box and scale it in the required directions.
2. One sphere should rotate counter clock wise around the y-axis (CCW) at a rate of 0.1
o per frame
(about 3o per second).
3. The second sphere should rotate around the y-axis clock wise (CW) at a rate of 0.2
o per frame (about
6
o per second).
4. The user can move the cylinder in the xz plane by using the i, j, k, l keyboard strokes –
(a) Pressing on i means translate the object by a small delta (e.g., 1) x along the x–axis = x –delta;
(b) Pressing on l means translate the object by a small delta (e.g., 1) along the x-axis x = x +
delta;
(c) Pressing on j means translate the object by a small delta (e.g., 1) along the z-axis z = z –delta;
(d) Pressing on k means translate the object by a small delta (e.g., 1) along the z-axis z = z +
delta;
4 (10 points) Part 4 – Render a "robot arm" complex hierarchical object (bonus)
Purpose:
• Create a more complex hierarchical object
• Use keyboard commands
4 COMP 3009
Figure 2: Simple hierarchical object
• Experiment with the simplicity of hierarchical objects
To do:
1. Create a simple "robot arm" (hierarchical object) as follows:
1.1. The object consists of 3 cylinders and 2 spheres (Figure 3).
2. The user can identify each of the parts using the numeric keys 1, 2, 3, 4, 5. Figure 2 shows how the
parts and their ids.
3. The user can manipulate the objects as follows (note all rotations are with respect to the model
object):
3.1. The user selects an object part using a numeric key 1-5
3.2. Action for cylindrical parts (id – 1, 3, 5)
3.2.1. Once a part is selected the user can rotate the object as follows: rotate the part using keys z
and x around the y-axis (yaw rotation) by +1 or -1 degree, respectively. See the orange arrows.
3.3. Action for spherical parts (id – 2, 4).
3.1. Once a part is selected the user can rotate the object as follows: rotate the part using keys z
and x around the x-axis (pitch rotation) by +1 or -1 degree, respectively. See the black arrow
4. Note that you may have to adjust the orientation of the box to ensure correct operation.
Figure 3: Robot arm manipulator
5 COMP 3009 | 677.169 | 1 |
How to measure curve distance in autocad?
Click Measure from the expanded Draw panel, or type ME. At the Select object to measure: prompt, click the end of the curve that you want to use as the starting point for your distance measurement. At the Specify length of segment or [Block]: prompt, enter the distance you want.
In this regard, how do you Measure the length of curved line?
People ask also, how do you Measuredistance in CAD?
Tap MEASURE.
Choose Distance.
Specify a first and a second point. Use object snaps for precision.
Additionally, how do you draw an exact arc length in Autocad? It could of course be done by calculating the angle mathematically, but an easier way to do it is to draw the arc with the required radius and centre point, then modify it using the lengthen tool on the Modify tab drop down. The arc will now be the desired length.
Beside above, how do you find the length of the curve between two points? If the arc is just a straight line between two points of coordinates (x1,y1), (x2,y2), its length can be found by the Pythagorean theorem: L = √ (∆x)2 + (∆y)2 , where ∆x = x2 − x1 and ∆y = y2 − y1.An opisometer, also called a curvimeter, meilograph, or map measurer, is an instrument for measuring the lengths of arbitrary curved lines.
Contents
How do you measure inclined lines in AutoCAD?
What is distance in AutoCAD?
The distance is displayed in the current units format. DIST assumes the current elevation for the first or second point if you omit the Z coordinate value. In paper space, distances are normally reported in 2D paper space units.
How do you draw an arc with radius and arc length in AutoCAD?
How do I use arc command in AutoCAD you draw a 7cm arc?
How do you find the length of the curve over the given interval?
The arc length of a curve y=f(x) over the interval [a,b] can be found by integration: ∫ba√1+[f′(x)]2dx.
How do you calculate a curve?
A simple method for curving grades is to add the same amount of points to each student's score. A common method: Find the difference between the highest grade in the class and the highest possible score and add that many points. If the highest percentage grade in the class was 88%, the difference is 12%.
How do you find the distance on a curved line on a map?
Use a ruler to measure the distance between the two places. If the line is quite curved, use a string to determine the distance, and then measure the string. Find the scale for the map you're going to use. It might be a ruler bar scale or a written scale, in words or numbers.
What is the instrument used to measure distance?
An odometer is a mechanical device that measures distance. An odometer wheel is used in surveying: this device counts the revolutions of a wheel and shows the distance traveled on a multiple dial readout (Figure 12.1).
How can you measure the length of a curved line for Class 6?
We take one end of the thread and put it on the point 'A' of the curved line. Now, we move the thread along the curved line and hold the other end of the thread when it reaches the point 'B' . Now, we stretch the thread along a meter ruler and measure it. This is the length of the curved line | 677.169 | 1 |
Background transparent Antialiasing
Caption top left top right bottom left bottom right none
Drwan angle
An angle describes how two intersecting straight lines (also called legs) are relative to each other. The larger the angle, the greater the distance between the lines as you move away from the point of intersection. This applies up to an angle of 90 or 180 degrees (°). At 90 degrees, the right angle, the distance is the greatest on both sides, at 180 degrees the straight lines are on top of each other, but in opposite directions.
If you continuously increase the angle from 0 to 360 degrees, then the legs describe a circular path. 180 degrees is a semicircle, 360 is a full circle. The division of the circle into 360 degrees is arbitrary and dates back to the ancient Babylonians, as are the divisions of a degree into 60 arc minutes and a arc minute into 60 arc seconds. Instead of minutes and seconds, you can also calculate in fractions of degrees, which is also common and a lot easier. A well-known example is the 23.5 degree tilt of the earth's axis to the plane of rotation, which causes our seasons.
Mathematically more obvious but more difficult to calculate than the degree is the radian, which refers to the circle number π (pi). A semicircle, i.e. 180 degrees, corresponds to one π, 2π is a full circle, i.e. 360 degrees, according to the calculation for the circumference of the circle as 2π times the radius. However, the various calculators on this page prefer to use the degree, because this is simply more catchy and well-known.
Usually, angles are denoted with lowercase Greek letters. In the case of a triangle, these are alpha (α), beta (β) and gamma (γ). In the case of a square, the delta (δ) is added, and so on. | 677.169 | 1 |
First principles of Euclid: an introduction to the study of the first book of Euclid's Elements
Αναζήτηση στο βιβλίο
Σελίδα 20 ... triangle is a figure ( ABC ) contained by three straight lines . ( b ) An equilateral tri- angle is that which is contained by three equal straight lines ( ABAC = BC ) . B C ( c ) Repeat the definition of a circle learned with the last ...
Σελίδα 32 ... triangle ABC . ( Euc . I. 1 shows how . ) ( b ) Bisect the angle A CB by the straight line CD , meet- ing A B at D. ( Euc . I. 9 shows how to do this . ) A D Then the line AB shall be bisected at the point D. If A B be bisected at D ...
Σελίδα 49 ... angles equal , each of them is a right angle . The line AB standing on the line CD makes the angle ABC equal to the angle A B D. .. Each of the angles ABC , ABD is a right angle . E ( b ) Let us assume that angle ABC is Theorem ( Euclid I.
Σελίδα 50 T S. Taylor. ( b ) Let us assume that angle ABC is not equal to angle ABD , as in this figure . For the purpose of proof draw EB at right angles to CD . In this figure we have now five angles . E D B Two angles CBA , ABD formed by the ...
Σελίδα 53 ... ABC , ABE . ( b ) ... The angles ABC , ABE are to- gether equal to two right angles . The angles ABC , ABD were ... angle ABC . ( The angle ABC being found in both pairs of angles , is called the common angle . ) Then the remaining angle ...
Σελίδα 18 66 34 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it.
Σελίδα 94 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of' the base, equal to one another, and likewise those which are terminated in the other extremity.
Σελίδα 104 - If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Σελίδα 51 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. | 677.169 | 1 |
Minor-axis Sentence Examples
If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.
0
0
If B and B' be points on the curve, BB' is the minor axis and C the centre of the curve.
0
0
An important metrical property of conjugate diameters is the sum of their squares equals the sum of the squares of the major and minor axis.
0
0
The major axis of the ellipse will be along the East-West line and the minor axis of the ellipse will be along the East-West line and the minor axis will be North-South.
0
0
The study has included the effects of initial imperfections, and has investigated the minor axis bending stresses that are induced.
0
0
Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the major axis of the ellipse, and the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the minor axis of the ellipse; about these axes the curve is symmetrical.
Related Articles
A line is defined as a one-dimensional geometric figure with length but no width. It extends infinitely in either direction with no ends, and the equation of a straight line is ax + b = 0. Keep reading to learn about the five main types of lines in geometry with line examples in your everyday life.
If you're studying linguistics or want to learn more about pronunciation, understanding the IPA vowel sounds chart is essential. The International Phonetic Alphabet (IPA) is a type of notation for various sounds used in language. The International Phonetic Association releases new IPA charts every few years to reflect current sounds and phonetic pronunciations. Understanding how the vowel chart works can be confusing, but this overview will help. | 677.169 | 1 |
4 Answers
4
You can calculate the vector spherical coordinates and then rotate the canvas using TikZ 3d library. I made a simple macro just to show the idea, but you can add more parameters as needed (center, radius, colors, etc.) or make a pic.
There are infinite circles as you say. In 3D, a circle is defined by a plane and a sphere. In this code, suppose you want to draw a circle with center at (a, b, c), radius rcircle, normal vector of the plane contains the circle is (a,b,c). The circle lies on the sphere with center at (a, b, c) and radius of sphere is rsphere=sqrt(a*a+b*b+c*c+rcircle*rcircle).
You can use 3dtools
3dtools can draw a circumcircle of a triangle. Therefore, if you want ta draw a circle on a plane knowing the given equation, you can choose three non-collinear points lies on this plane.
This code draw a circle on the plane has the equation 2 x + 2 y - z = 7 | 677.169 | 1 |
Important questions for Class 10 Maths Chapter 9 Some Applications of Trigonometry are provided here for the board exams preparation. The questions are based on the new pattern of CBSE and are as per the revised 2021 syllabus. Students who are preparing CBSE 2021 Maths exam are advised to practice these important questions of Some Applications of Trigonometry For Class 10. Solving these questions will help students to score high marks in the questions asked from this chapter.
Trigonometry has more applications in our daily existence, and hence, the chapter is not only important for the board exam but also useful in many other fields. Most of the questions from this chapter are also asked in the competitive exams such as JEE etc.
Below, we have provided the questions of Chapter 9 Applications of Trigonometry with the solutions. Students can also find some questions without solutions for their practice.
Q.1: The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it is 60°. Find the height of the tower.
Solution:
Let AB be the tower and BC be the length of its shadow when sun's altitude (angle of elevation from the top of the tower to the tip of the shadow) is 60° and DB be the length of the shadow when the angle of elevation is 30°.
Let us assume, AB = h m, BC = x m
DB = (40 +x) m
In right triangle ABC,
tan 60° = AB/BC
√3 = h/x
h = √3 x……….(i)
In right triangle ABD,
tan 30° = AB/BD
1/√3 =h/(x + 40) ……..(ii)
From (i) and (ii),
x(√3 )(√3 ) = x + 40
3x = x + 40
2x = 40
x = 20
Substituting x = 20 in (i),
h = 20√3
Hence, the height of the tower is 20√3 m.
Q. 2:A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
Solution:
Using given instructions, draw a figure. Let AC be the broken part of the tree. Angel C = 30 degrees.
BC = 8 m
To Find: Height of the tree, which is AB
From figure: Total height of the tree is the sum of AB and AC i.e. AB+AC
In right ΔABC,
Using Cosine and tangent angles,
cos 30° = BC/AC
We know that, cos 30° = √3/2
√3/2 = 8/AC
AC = 16/√3 …(1)
Also,
tan 30° = AB/BC
1/√3 = AB/8
AB = 8/√3 ….(2)
From (1) and (2),
Total height of the tree = AB + AC = 16/√3 + 8/√3 = 24/√3 = 8√3 m.
Q. 3: Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.
Solution:
Let AB and CD be the poles of equal height.
O is the point between them from where the height of elevation taken. BD is the distance between the poles.
As per the above figure, AB = CD,
OB + OD = 80 m
Now,
In right ΔCDO,
tan 30° = CD/OD
1/√3 = CD/OD
CD = OD/√3 … (1)
In right ΔABO,
tan 60° = AB/OB
√3 = AB/(80-OD)
AB = √3(80-OD)
AB = CD (Given)
√3(80-OD) = OD/√3 (Using equation (1))
3(80-OD) = OD
240 – 3 OD = OD
4 OD = 240
OD = 60
Substituting the value of OD in equation (1)
CD = OD/√3
CD = 60/√3
CD = 20√3 m
Also,
OB + OD = 80 m
⇒ OB = (80-60) m = 20 m
Therefore, the height of the poles are 20√3 m and distance from the point of elevation are 20 m and 60 m respectively.
Q. 4:A pole 6 m high casts a shadow 2√ 3 m long on the ground, then the Sun's elevation is
(A) 60° (B) 45° (C) 30° (D) 90°
Solution:
Let BC be the pole and AB be its shadow.
In triangle ABC,
tan θ = BC/AB
= 6/2√ 3
= 3/√ 3
= (√ 3 × √ 3 )/ √ 3
= √ 3
tan θ = tan 60°
θ = 60°
Hence, the Sun's elevation is 60°.
Q. 5: The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is √st.
Solution:
Let BC = s; PC = t
Let the height of the tower be AB = h.
∠ABC = θ and ∠APC = 90° – θ
(∵ the angle of elevation of the top of the tower from two points P and B are complementary)
The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of top of the tower from the foot of the hill is 30°.If the tower is 50 m high, what is the height of the hill?
A bridge across the river makes an angle of 45° with the river bank. If the length of the bridge across the river is 150 m, what is the width of the river?
There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. P and Q are the points directly opposite to each other on the two banks, and in a line with the tree. If the angles of elevation of the top of tree from P and Q are 30° and 45° respectively, find the height of the tree.
From a point 20m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower.
A flagstaff stands at the top of 5m high tower. From a point on the ground, the angle of elevation of the top of the flagstaff is 60° and from the same point, the angle of elevation of the top of the tower is 45°. Find the height of the flagstaff.
A tower subtends an angle α at a point A in the place of its base and the angle of depression of the foot of the tower at a point b ft. just above A is β. Prove that the height of the tower is b tan α cot β.
A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. From a point on the plane, the angle of elevation of the bottom of the flagstaff is 30° and that of the top of the flagstaff is 45°. Find the height of the tower | 677.169 | 1 |
Find a vector $\overrightarrow{\mathrm{d}}$ which is perpendicular to both $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, and is such that $\overrightarrow{\mathrm{d}} \cdot \overrightarrow{\mathrm{c}}=21$. | 677.169 | 1 |
Unlocking the Power of Triangles: How to Use Them in Your Daily Life
Triangles are one of the most fundamental shapes in geometry and have significant importance in various fields. From mathematics to design, triangles play a crucial role in understanding and solving complex problems. They are not only a basic shape but also possess unique properties that make them versatile and powerful. In this article, we will explore the different aspects of triangles and their significance in different areas of study and practice.
Key Takeaways
Triangles are a basic geometric shape with three sides and three angles.
Triangles can be used as a tool to achieve goals and create eye-catching designs.
Triangles can simplify complex equations in mathematics and help explore shape and space in geometry.
Triangles play a role in physics by understanding force and motion.
Triangles have spiritual significance and can be used in meditation and healing.
Understanding the Basics: What are Triangles and How Do They Work?
A triangle is a polygon with three sides and three angles. It is the simplest polygon and serves as the building block for more complex shapes. Triangles have several basic properties that make them distinct. Firstly, the sum of the interior angles of a triangle is always 180 degrees. This property allows for easy calculation of unknown angles in a triangle. Secondly, the lengths of the sides of a triangle determine its shape and classification.
There are several types of triangles based on their side lengths and angle measurements. An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees. An isosceles triangle has two equal sides and two equal angles. A scalene triangle has no equal sides or angles. Triangles can also be classified based on their angles, such as acute, obtuse, or right triangles.
The Pythagorean theorem is a fundamental concept in geometry that relates to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem has wide applications in solving triangle problems, such as finding missing side lengths or angles.
The Power of Triangles: How They Can Help You Achieve Your Goals
The concept of the "triangle of success" is a powerful tool for goal-setting and achievement. The triangle represents three key elements necessary for success: passion, skills, and opportunity. Passion refers to the deep desire and enthusiasm for a particular goal or endeavor. Skills represent the knowledge, abilities, and expertise required to excel in that field. Opportunity refers to the favorable circumstances or situations that can help in achieving the goal.
By visualizing these three elements as the vertices of a triangle, individuals can gain clarity and focus on their goals. The triangle serves as a reminder of the interplay between passion, skills, and opportunity. It helps individuals identify areas where they need to develop their skills or seek new opportunities to align with their passion. By consciously working on each element of the triangle, individuals can increase their chances of achieving their goals.
Triangles can also be used as a visual aid in achieving personal and professional goals. By creating a visual representation of their goals using triangles, individuals can gain a better understanding of the different components and how they relate to each other. For example, they can create a triangle with each side representing a different aspect of their goal, such as knowledge, experience, and networking. By visually tracking their progress in each area, individuals can stay motivated and focused on their goals.
Using Triangles in Design: Tips for Creating Eye-Catching Graphics
Design Element
Description
Benefits
Triangles
Geometric shape with three sides and three angles
Can add visual interest and depth to a design, can create a sense of movement and direction
Color
The hue, saturation, and brightness of a design element
Can be used to create contrast and emphasis, can evoke emotions and set a mood
Composition
The arrangement of design elements within a space
Can create balance and harmony, can guide the viewer's eye through the design
Typography
The style, size, and arrangement of text in a design
Can add personality and tone to a design, can improve readability and hierarchy
Negative Space
The empty space around and between design elements
Can create contrast and balance, can improve readability and focus
Triangles are widely used in graphic design to create dynamic and visually appealing compositions. They add energy and movement to a design and can be used to direct the viewer's attention. Here are some tips for using triangles effectively in design:
1. Use triangles as framing elements: Triangles can be used to frame important elements in a design, such as text or images. By placing these elements within a triangular shape, designers can draw attention to them and create a focal point.
2. Create balance with triangles: Triangles can be used to create balance in a design by distributing visual weight evenly. By placing triangles of different sizes or orientations on opposite sides of a composition, designers can achieve a sense of equilibrium.
3. Use triangles to create depth and perspective: Triangles can be used to create the illusion of depth and perspective in a design. By overlapping or intersecting triangles, designers can create a sense of three-dimensionality.
4. Experiment with different triangle orientations: Triangles can be oriented in different ways to create different effects. For example, an upward-pointing triangle can convey stability and strength, while a downward-pointing triangle can suggest movement or change.
5. Combine triangles with other shapes: Triangles can be combined with other shapes to create more complex compositions. By combining triangles with circles, squares, or other polygons, designers can create visually interesting and dynamic designs.
Triangles in Mathematics: How They Can Simplify Complex Equations
Triangles play a crucial role in trigonometry, a branch of mathematics that deals with the relationships between the angles and sides of triangles. Trigonometry is widely used in fields such as physics, engineering, and navigation to solve complex equations and problems.
One of the key concepts in trigonometry is the trigonometric functions: sine, cosine, and tangent. These functions relate the angles of a triangle to the ratios of its sides. For example, the sine function relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
Trigonometry allows for the calculation of unknown angles or side lengths in a triangle based on known information. This is particularly useful in real-world applications where measurements may be incomplete or inaccurate. By using trigonometric functions and the properties of triangles, complex equations involving angles and distances can be simplified and solved.
Triangles also have a close relationship with circles in geometry. The unit circle is a circle with a radius of 1 unit that is centered at the origin of a coordinate plane. By placing a right triangle within the unit circle, trigonometric functions can be defined and related to the coordinates of points on the circle. This relationship between triangles and circles allows for the application of trigonometry in solving problems involving circular motion, such as calculating the position or velocity of an object moving in a circular path.
The Role of Triangles in Geometry: Exploring Shape and Space
Triangles have significant importance in geometry and serve as the foundation for understanding shape and space. They are the simplest polygon and possess unique properties that make them versatile and powerful tools in geometry.
Triangles are used to calculate the area and perimeter of shapes. The area of a triangle can be calculated using the formula A = 1/2 * base * height, where the base is the length of one side of the triangle and the height is the perpendicular distance from the base to the opposite vertex. The perimeter of a triangle is simply the sum of the lengths of its three sides.
Triangles also have relationships with other shapes in geometry. For example, any polygon can be divided into triangles, allowing for easier calculation of its area or perimeter. Triangles are also used to classify other polygons based on their properties. For instance, a quadrilateral can be classified as a parallelogram if its opposite sides are parallel, which can be proven by dividing it into two triangles.
The concept of similarity is another important aspect of triangles in geometry. Two triangles are said to be similar if their corresponding angles are congruent and their corresponding sides are proportional. Similar triangles have several properties that allow for easier calculation and comparison of their dimensions.
Triangles in Physics: Understanding Force and Motion
Triangles play a crucial role in physics, particularly in understanding force and motion. They are used to calculate vectors, which represent quantities that have both magnitude and direction.
In physics, vectors are often represented by arrows, with the length of the arrow representing the magnitude of the vector and the direction of the arrow representing its direction. By using triangles, vectors can be broken down into their components, which are vectors in the x and y directions.
The Pythagorean theorem is used to calculate the magnitude of a vector by combining its x and y components. The magnitude of a vector is equal to the square root of the sum of the squares of its components. This allows for the calculation of the total force or velocity of an object based on its individual components.
Triangles are also used to calculate forces in physics. When multiple forces act on an object, they can be represented by vectors. By using triangles, these vectors can be added or subtracted to determine the net force acting on the object. The net force determines the acceleration or motion of the object according to Newton's second law of motion.
The Spiritual Significance of Triangles: Symbolism and Meaning
Triangles have deep symbolism and meaning in various spiritual and religious traditions. They are often associated with balance, harmony, and unity. In many cultures, triangles are seen as sacred symbols that represent the divine trinity or the union of mind, body, and spirit.
In Christianity, the triangle is often used to represent the Holy Trinity – God the Father, Son, and Holy Spirit. It symbolizes the unity and interconnectedness of these three aspects of God. In Hinduism, the triangle represents Shakti, the divine feminine energy that is believed to create and sustain the universe.
In esoteric traditions such as Freemasonry and alchemy, triangles are used as symbols of transformation and spiritual enlightenment. The upward-pointing triangle represents fire or spirit, while the downward-pointing triangle represents water or matter. The combination of these two triangles creates a hexagram or Star of David, which symbolizes the union of opposites and the balance between masculine and feminine energies.
Harnessing the Power of Sacred Geometry: Using Triangles in Meditation and Healing
Sacred geometry is a concept that explores the relationship between geometric shapes and spiritual or metaphysical principles. Triangles are an integral part of sacred geometry and are believed to have healing and transformative properties.
In meditation practices, triangles are often used as focal points for concentration and visualization. By visualizing a triangle in the mind's eye, individuals can create a sense of balance and harmony within themselves. Triangles are also used as symbols of protection and grounding in meditation practices.
In energy healing modalities such as Reiki or crystal healing, triangles are used to direct and amplify energy. By placing crystals or other healing tools in the shape of a triangle, practitioners can enhance the flow of energy and promote healing. Triangles are also used in sacred geometry grids, where specific crystals or symbols are arranged in geometric patterns to create a harmonious energy field.
Triangles in Nature: Discovering Patterns and Symmetry in the Natural World
Triangles can be found abundantly in nature, from the structure of crystals to the patterns on butterfly wings. They are often associated with strength, stability, and efficiency.
One of the most famous examples of triangles in nature is the honeycomb structure created by bees. Bees construct hexagonal cells that fit together to form a honeycomb. These hexagons are made up of smaller triangles, which provide strength and stability to the structure. The triangular shape allows for efficient use of space and materials.
Another example of triangles in nature is found in the formation of snowflakes. Snowflakes are formed when water vapor freezes into ice crystals. The intricate patterns on snowflakes are based on hexagonal symmetry, which is created by the arrangement of triangular ice crystals.
Triangles can also be found in the structure of leaves, flowers, and even human bodies. The branching patterns of trees often follow a triangular shape, allowing for efficient distribution of nutrients and sunlight. In human anatomy, triangles can be seen in the structure of bones, muscles, and organs, providing strength and stability to the body.
Applying Triangles in Everyday Life: Tips for Using Them in Home Decor and Fashion
Triangles can be applied in everyday life to create a modern and stylish look in home decor and fashion. Here are some practical tips for using triangles in these areas:
1. Home Decor: Triangles can be used in home decor to create a contemporary and geometric look. They can be incorporated into wallpaper, rugs, or artwork to add visual interest and depth to a space. Triangular shelves or bookcases can also be used to display decorative items or books.
2. Fashion: Triangles can be used in fashion to create bold and eye-catching designs. They can be incorporated into clothing patterns, accessories, or jewelry to add geometric elements to an outfit. Triangular shapes can also be used in hairstyles or makeup to create unique and edgy looks.
3. Graphic Prints: Triangular patterns can be used in graphic prints on clothing or home decor items. These prints can range from simple geometric designs to more intricate patterns inspired by nature or cultural motifs. Triangular prints can add a modern and dynamic touch to any outfit or space.
4. Color Blocking: Triangles can be used in color blocking techniques to create interesting visual effects. By combining different colors in triangular shapes, designers can create a bold and vibrant look. This technique can be applied to clothing, accessories, or home decor items.
5. Jewelry Design: Triangular shapes can be used in jewelry design to create unique and geometric pieces. Triangular gemstones or metal accents can add a modern and edgy touch to necklaces, earrings, or bracelets.
Triangles are not just simple shapes; they have significant importance in various fields of study and practice. From mathematics to design, triangles play a crucial role in understanding complex problems and achieving personal and professional goals. They are versatile and powerful tools that can simplify equations, create visually appealing compositions, and symbolize balance and harmony.
Understanding the basics of triangles, such as their properties and classifications, is essential for building a strong foundation in geometry and trigonometry. By harnessing the power of triangles, individuals can set clear goals, visualize their progress, and achieve success in their endeavors.
Triangles also have spiritual significance and can be used in meditation and healing practices to create balance and promote well-being. In nature, triangles can be found in various structures and patterns, showcasing their efficiency and strength.
In everyday life, triangles can be applied in home decor and fashion to create a modern and stylish look. By incorporating triangular shapes into designs, individuals can add visual interest and depth to their surroundings.
In conclusion, triangles are not just simple shapes; they are powerful tools that have the potential to transform our understanding of the world around us. By exploring the different aspects of triangles and their significance in various fields, we can unlock their potential for personal and professional growth | 677.169 | 1 |
1-1 Points Lines And Planes Worksheet Answers
Exclusive worksheets on planes include collinear and coplanar concepts. Points lines and planes gina wilson answer key displaying top 8 worksheets found for this concept.
Point Of Concurrency Triangle Worksheet Plane Geometry Circle Diagram
Displaying top 8 worksheets found for naming points lines and planes practice.
1-1 points lines and planes worksheet answers. If points on are the same plane they are coplanar. Consider two lines in a plane. Some of the worksheets for this concept are unit 1 tools of geometry reasoning and proof identify points lines and planes points lines and planes exercise 1 use the figure to name each of the geometry unit 1 workbook geometry lesson points lines and planes lines and angles name practice identify.
Use the figure to name each of the following. 2 4 use postulates and diagrams obj. Section 1 1 worksheet 4 understanding points lines and planes lines in a plane divide the plane into regions.
Use postulates involving points lines and planes. The number of regions depends on the relationship between the lines. A line is a collection of points along a straight path with no end points.
If the lines are if the lines parallel three intersect four regions are regions are determined. A ray has one dimension. A point is a location determined by an ordered set of coordinates.
If points are on the same line they are collinear. This ensemble of printable worksheets for grade 8 and high school contains exercises to identify and draw the points lines and planes. Some of the worksheets for this concept are points lines and planes exercise 1 points lines and planes identify points lines and planes chapter 4 lesson1 0 points line segments lines and rays points lines planes angles 1 3 points lines and planes clinton public school district clinton public schools 11.
Points lines and planes gina wilson displaying top 8 worksheets found for this concept. Interesting descriptive charts multiple choice questions and word problems are included in these pdf worksheets. A line has one dimension.
Points lines and planes point a point has zero dimensions. Some of the worksheets for this concept are identify points lines and planes work section 3 1 parallel lines and transversals use the figure to name each of the unit 1 tools of geometry reasoning and proof the segment addition postulate date period geometry unit 1 workbook finding. A plane exists in two dimensions.
Key vocabulary line perpendicular to a plane a line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. Postulate in geometry rules that are accepted without proof are. Name points lines and planes in geometry a point is a location a line contains points and a plane is a flat surface that contains points and lines.
Some of the worksheets for this concept are points lines and planes exercise 1 points lines and planes 1 chapter 1 lesson 1 points and lines in the plane identify points lines and planes 3 points in the coordinate name answer key points lines and planes 1 chapter 4 lesson1 0 points line segments lines and rays. A line segment has one dimension. Displaying top 8 worksheets found for plane line | 677.169 | 1 |
A 6-sided figure with all its sides and angles equal is called a regular hexagon: How many lines of symmetry does a regular hexagon have?
A
3
No worries! We've got your back. Try BYJU'S free classes today!
B
6
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
9
No worries! We've got your back. Try BYJU'S free classes today!
D
12
No worries! We've got your back. Try BYJU'S free classes today!
Open in App
Solution
The correct option is B 6
A regular hexagon is said to have six lines of symmetry, 3 joining the opposite vertices and 3 joining the mid-points of the opposite sides. ( Note: A regular polygon of "n" sides has "n" Lines of Symmetry ). | 677.169 | 1 |
FAQs on Perimeter of a Trapezoid Formula
1. Is the trapezoid/trapezium considered a quadrilateral?
A quadrilateral is a closed shape with four sides, as we all know. As a result, a trapezoid/trapezium is sometimes called a quadrilateral. Some people describe a trapezoid as a quadrilateral with just one set of parallel sides (the exclusive definition), eliminating parallelograms from the definition. Others define a trapezoid as a quadrilateral having at least one pair of parallel sides (inclusive definition), which makes the parallelogram a specific sort of trapezoid. Angles of different dimensions can be found on a trapezoid.
2. Is there a difference between a trapezoid and a trapezium?
A trapezium is a quadrilateral with one set of opposing parallel sides. A trapezium's parallel sides are known as bases, and its non-parallel sides are known as legs. It's also known as a trapezoid. A trapezoid with two parallel sides is another name for a parallelogram.
A trapezium is a four-sided flat form with straight sides and opposing sides that are parallel. In the United States, it's known as a trapezoid.
A trapezoid is a quadrilateral with NONE of its sides parallel to one other.
3. What are the three names for a trapezoid?
The Scalene Trapezoid is a trapezoid with no equal-sided sides.
The two opposing sides of an isosceles trapezoid are parallel, while the two other sides are of equal length. In addition, the diagonals are of the same length.
A trapezoid with a right angle is known as a right trapezoid. Properties. It's at a 45-degree angle.
Obtuse Trapezoid: There must be two obtuse angles in the trapezoid. One of the legs is perpendicular to the foundations.
Sharp Trapezoid: On its longer base edge, it features two consecutive acute angles. It's a trapezoid with the same measure of base angles.
4. How to calculate the area of a trapezoid?
There are two approaches to prove the formula for finding the area of a trapezoid:
Using a parallelogram as proof
Using a triangle as a proof
Using a triangle, we'll demonstrate how to prove the area of a trapezoid formula. Consider the trapezoid with the bases a and b and the height h shown above. To demonstrate the formula,
One of the legs should be divided into two halves.
A triangular section of the trapezoid should be cut off.
Attach it to the bottom of the figure.
The trapezoid is reorganized into a triangle in this manner. The areas of the trapezoid and the triangle are the same, as we can see in the picture above. We can also observe that the triangle's base is (a + b) and that the triangle's height is h. | 677.169 | 1 |
The Elements of Spherical Trigonometry
From inside the book
Results 1-5 of 13
Page 4 ... angles must be greater than the third . 6. Since the solid angle at O ( see fig . p . 3 ) is contained by three ... opposite angles and sides of the other , and vice versa . Since B is the pole of D F , then B D is a quadrant , and since ...
Page 6 ... angles are to each other as the sines of their opposite sides . 14. 3rd , Relation between the two sides and their included angle , aud the angle opposite one of them . In considering the combination a , b , A , C ; first eliminate cos ...
Page 10 ... angles , if we make any one of the angles = 90 ¯ . If A = 90 ¯ we have cos a = cos b cos c ...... ( 1 ) sin b = sin ... opposite ; the third , between the hypothenuse , a side , and the adjacent angle ; the fourth , between the two sides and | 677.169 | 1 |
0 users composing answers..
A diagram would be of great assistance for this problem, so I created one for you. I added lines and points where I saw fit to add clarity to the diagram:
In order to find the area of any triangle, we need the length of the base and the length of the perpendicular height (also known as the altitude) of the triangle. Then, we can apply the formula \(A_{\triangle ABC}=\frac{1}{2}bh\) to find the area. In the diagram above, \(\overline{AB}\) represents the base, and \(\overline{EC}\) represents the perpendicular height of the triangle.
\(m\angle ADB=120^{\circ}\)as each central angle \(\triangle ABC\) forms divides equally in thirds. By inserting an angle bisector of \(\angle ADB\), this means that \(m\angle ADE=60^{\circ}\). This angle bisector also happens to be a perpendicular of chord \(\overline{AB}\). Combining all of this information together means that \(\triangle ADE\) is a 30-60-90 triangle.
A 30-60-90 triangle has a constant ratio of its sides of \(1:\sqrt{3}:2\). In this particular case, \(r\) is the length of the hypotenuse of \(\triangle ADE\). Let's use these ratios to find the length of the base and the height in terms of \(r\).
\(\frac{AE}{r}=\frac{\sqrt{3}}{2}\\ AE=\frac{\sqrt{3}}{2}r\)
\(\frac{DE}{r}=\frac{1}{2}\\ DE=\frac{1}{2}r\)
By parallel reasoning, \(BE=\frac{\sqrt{3}}{2}r\).
We can find \(AB\), the length of the base of the triangle in terms of r now. | 677.169 | 1 |
Lines Of Symmetry Does A Pentagon Have?
How Many Lines Of Symmetry Does A Octagon Have Nakley from nakley.me
Introduction
Symmetry is an incredibly important concept in mathematics. It is a fundamental part of geometry, and it has applications in many other areas like physics and engineering. One of the most basic shapes that exhibits symmetry is the pentagon, a five-sided polygon. But how many lines of symmetry does a pentagon have?
What is a Pentagon?
A pentagon is a two-dimensional shape with five sides. It is one of the most basic shapes in geometry, and it is used to construct many different shapes, like pentagrams and pentagons. It is also a regular polygon, which means that it has all of its sides equal in length and all of its angles equal in measure.
What is Symmetry?
Symmetry is a concept in mathematics that describes the equality of shapes or objects. It is a way of describing how a shape or object is the same on both sides, or how it looks the same when it is rotated or reflected. Symmetry is an important part of geometry, and it is used to describe the properties of shapes and objects in many different fields.
How Many Lines of Symmetry Does a Pentagon Have?
A pentagon has five lines of symmetry. This means that if you draw a line through the center of the pentagon, it will be divided into two equal halves. The same is true if you draw a line through any of the sides of the pentagon. Each side will be divided into two equal halves, creating five lines of symmetry.
Why Does a Pentagon Have Five Lines of Symmetry?
A pentagon has five sides, so it makes sense that it would have five lines of symmetry. The lines of symmetry divide the pentagon into two equal halves, which is a property of regular polygons in general. The number of lines of symmetry in a regular polygon is always equal to the number of sides of the polygon.
What Other Shapes Have Five Lines of Symmetry?
In addition to the pentagon, there are several other shapes that have five lines of symmetry. These shapes include the pentagram, the octagon, and the decagon. All of these shapes are regular polygons, which means that all of their sides are equal in length and all of their angles are equal in measure.
Conclusion
A pentagon is a five-sided regular polygon that has five lines of symmetry. The lines of symmetry divide the pentagon into two equal halves, and this is a property of regular polygons in general. There are several other shapes that have five lines of symmetry, including the pentagram, the octagon, and the decagon.
Summary
Symmetry is an important concept in mathematics, and one of the most basic shapes that exhibits symmetry is the pentagon. The pentagon has five lines of symmetry, which means that if you draw a line through the center of the pentagon, it will be divided into two equal halves. The same is true if you draw a line through any of the sides of the pentagon. There are several other shapes that also have five lines of symmetry, including the pentagram, the octagon, and the decagon. | 677.169 | 1 |
The Elements of Euclid; viz. the first six books, together with the eleventh ...
Let AB be the given straight line; It is required to describe a square upon AB.
E
From the point A drawa AC at right angles to AB; and 3. 1. make AD equal to AB, and through the point D draw DE 31. 1. parallel to AB, and through B draw BE parallel to AD; 34. 1. therefore ADEB is a parallelogram: whence AB is equald to DE, and AD to BE: but BA is equal to AD; therefore the four straight lines BA, AD, DE, EB, are equal to one another, and the parallelogram ADEB is equilateral, like- D wise all its angles are right angles; because the straight line AD meeting the parallels AB, DE, the angles € 29. 1. BAD, ADE are equale to two right angles: but BAD is a right angle; therefore also ADE is a right angle ; A but the opposite angles of parallelograms are equald; therefore each of the opposite angles ABE, BED is a right angle; wherefore the figure ADEB is rectangular, and it has been demonstrated that it is equilateral; it is therefore a square, and it is described upon the given straight line AB: Which was to be done.
B
COR. Hence every parallelogram that has one right angle has all its angles right angles.
46. 1.
PROP. XLVII. THEOR.
IN any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
Let ABC be a right-angled triangle having the right angle BAC; the square described upon the side BC is equal to the squares described upon BA, AC.
On BC describes the square BDEC, and on BA, AC the
squares GB, HC; and through A drawb AL parallel to BD, Book I. or CE, and join AD, FC. Then, because each of the angles
BAC, BAG is a rightangle, the two straight lines AC, AG, upon the opposite sides of AB, make with it at the point A the adjacent angles equal to two right angles; therefore CA is in the same straight lined with AG; for the same reason, AB and AH are in the same straight line; and because the angle DBC is equal to the angle FBA, each of them being a
angle DBA is equale to the whole FBC; and because the * 2 Ax. two sides AB, BD, are equal to the two FB, BC, each to each, and the angle DBA equal to the angle FBC; therefore the base AD is equalf to the base FC, and the triangle 4. 1. ABD to the triangle FBC: Now the parallelogram BL is doubles of the triangle ABD, because they are upon the 41. 1. same base BD, and between the same parallels BĎ, AL; and the square GB is double of the triangle FBC, because these also are upon the same base FB, and between the same parallels FB, GC. But the doubles of equals are equal to 6 Ax. one another: Therefore the parallelogram BL is equal to the square GB: And, in the same manner, by joining AE, BK it is demonstrated, that the parallelogram CL is equal to the square HC; Therefore the whole square BDEC is equal to the two squares GB, HC; and the square BDEC is described upon the straight line BC, and the squares GB, HC upon BA, AC: Wherefore the square upon the side BC is equal to the squares upon the sides BA, AC. Therefore, in any right-angled triangle, &c. Q. É. D.
PROP. XLVIII. THEOR.
If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.
BOOK I.
a 11. 1.
If the
square described upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other sides BA, AC, the angle BAC is a right angle.
D
From the point A drawa AD at right angles to AC, and make AD equal to BA, and join DC: Then, because DA is equal to AB, the square of DA is equal to the square of AB: To each of these add the square of AC; therefore the squares of DA, AC are equal to the squares of BA, AC: But the square of 47. 1. DC is equal to the squares of DA, AC,
because DAC is a right angle; and the square of BC, by hypothesis, is equal to the squares of BA, AC; therefore the square of DC is equal to the square of B
A
BC; and therefore also the side DC is equal to the side BC. And because the side DA is equal to AB, and AC common to the two triangles DAC, BAC, the two DA, AC are equal to the two BA, AC; and the base DC is equal 8. 1. to the base BC; therefore the angle DAC is equal to the angle BAC; but DAC is a right angle; therefore also BAC is a right angle. Therefore, if the square, &c. Q. E. D.
THE
ELEMENTS
OF
43
EUCLI D.
BOOK II.
DEFINITIONS.
I.
EVERY right-angled parallelogram is said to be Book II. contained by any two of the straight lines which contain one of the right angles.
II.
In every parallelogram, any of the parallelograms about
a diameter, together with A
Thus
the two complements, is called a Gnomon. 'the parallelogram HG, 'together with the com- 'plements AF, FC, is the gnomon, which is more 'briefly expressed by the
letters AGK, or EHC,
D
K
C
'which are at the opposite angles of the parallelograms 'which make the gnomon."
PROP. I. THEOR.
If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines, is equal to the rectangles contained by the undivided line, and the several parts of the divided line.
BOOK II.
Let A and BC be two straight lines; and let BC be divided into any parts in the points D, E; the rectangle con
tained by the straight lines A, B
BC is equal to the rectangle
contained by A, BD, together with that contained by A, DE, and that contained by A, EC. From the point B drawa G BF at right angles to BC, 3. 1. and make BG equal to A; 31. 1. and through Gdraw GH pa- F
a 11. 1.
rallel to BC; and through D, E, C, draw DK, EL, CH parallel to BG; then the rectangle BH is equal to the rectangles BK, DL, EH; and BH is contained by A, BC, for it is contained by GB, BC, and GB is equal to A; and BK is contained by A, BD, for it is contained by GB, BD, of which GB is equal to A; and DL is contained by A, DE, 34. 1. because DK, that isd BG is equal to A; and in like manner the rectangle EH is contained by A, EC: Therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, and by A, DE: and also by A, EC. Wherefore, if there be two straight lines, &c. Q. E. D.
a 46. 1.
b
PROP. II. THEOR.
IF a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line.
Let the straight line AB be divided A into any two parts in the point C; the rectangle contained by AB, BC, toge- ther with the rectangle* AB, AC, shall be equal to the square of AB.
B
Upon AB describe the square 31. 1. ADEB, and through C drawb CF parallel to AD or BE. Then AE is equal to the rectangles AF, CE; and AE is the square of AB; and AF is the rectangle contained by BA, AC; for it is contained by DA, AC of which AD is equal to AB; and CE is contain | 677.169 | 1 |
PYTHAG - The Right-Angled Triangle
What is the hypotenuse?
It is important for a lot of the work we are about to do in trigonometry that you can identify the hypotenuse of any right-angled triangle. Like we have in these ones here.
Pythagoras' Theorem
A long time ago, a Greek mathematician named Pythagoras came up with an interesting observation about right-angled triangles: if $c$c represents the length of the hypotenuse, and $a$a and $b$b represent the other two sides (that meet at a right angle), then $a^2+b^2=c^2$a2+b2=c2.
In other words, the square of the hypotenuse ($c$c) of a right-angled triangle is equal to the square of side $a$a plus the square of side $b$b.
The following interactive demonstrates using areas Pythagoras' Theorem. You can use the slider and move the squares of a and b to cover the area of c squared, thus demonstrating that $a^2+b^2=c^2$a2+b2=c2
To see how such a simple formula can be used to solve real world problems, suppose a carpenter wants to build a ramp to replace the set of stairs to his front porch. If the porch is $1$1 metre off the ground and the ramp is to start $4$4 metres from the base of the porch (in order to connect the ramp to the driveway), what is the length of the ramp he has to build?
To find out, we substitute the numbers into the formula: $c^2=1^2+4^2$c2=12+42, so $c^2=17$c2=17. Taking the square root of both sides, $c=\sqrt{17}$c=√17 so the ramp should be about $4.12$4.12 metres long.
Pythagoras' Theorem
$a^2+b^2=c^2$a2+b2=c2
for all right-angled triangles, where $c$c is the hypotenuse of the triangle.
So to test if a triangle is right-angled, we can use the Pythagorean Formula and the lengths of the sides. If the formula holds true, it is a right-angled triangle.
Example
Question 1
Which side of the triangle in the diagram is the hypotenuse?
$AB$AB
A
$CA$CA
B
$BC$BC
C
Question 2
Use Pythagoras' theorem to determine whether this is a right-angled triangle.
A triangle with labeled side lengths. The triangle has sides with different lengths and with no implicit indication of angle measurements. The sides are measured with lengths of $10$10 units, $18$18 units and $21$21 units.
Let $a$a and $b$b represent the two shorter side lengths. First find the value of $a^2+b^2$a2+b2.
Let $c$c represent the length of the longest side. Find the value of $c^2$c2.
Is the triangle a right-angled triangle?
Yes
A
No
B
Question 3
Is a triangle with side lengths of $3$3 cm, $4$4 cm and $5$5 cm a right-angled triangle?
Think: I will need to identify the longest side, this value will be c in the Pythagorean formula. | 677.169 | 1 |
-b) formula, which allows us to express the sine of the difference of two angles in terms of the sines and cosines of those angles. In this article, we will explore the sin(a-b) formula in detail, understand its derivation, and examine its practical applications.
Understanding Trigonometric Identities
Trigonometric identities are equations that relate the trigonometric functions (sine, cosine, tangent, etc.) of an angle to each other. These identities are derived from the geometric properties of triangles and are essential tools in solving trigonometric equations and simplifying expressions. The sin(a-b) formula is one such identity that helps us express the sine of the difference of two angles in terms of the sines and cosines of those angles.
The Sin(a-b) Formula
The sin(a-b) formula states that:
sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
This formula can be derived using the sum-to-product identities, which are a set of trigonometric identities that express the sum or difference of two trigonometric functions in terms of their products. The derivation involves manipulating the sum-to-product identities and applying basic algebraic principles. While the derivation itself may be complex, the sin(a-b) formula is relatively straightforward to use once understood.
Applying the Sin(a-b) Formula
The sin(a-b) formula finds applications in various areas, including physics, engineering, and geometry. Let's explore a few practical examples to understand how this formula can be used.
Example 1: Calculating the Sine of the Difference of Two Angles
Suppose we want to find the value of sin(60° – 30°). Using the sin(a-b) formula, we can express this as:
sin(60° – 30°) = sin(60°)cos(30°) – cos(60°)sin(30°)
By substituting the known values of sin(60°), cos(30°), cos(60°), and sin(30°) into the formula, we can calculate the result: | 677.169 | 1 |
Parallel lines – Simple geometry
Two lines are parallel if they are in the same plane but they can go on forever without ever crossing. You can test this by seeing if you can draw a third line that crosses both lines at right angles (it is perpendicular to them).
Parallel lines can be close together or far apart, but if you connect them with a lot of perpendicular lines (like a ladder), all of the rungs of the ladder will be the same length.
Two parallel lines connected by perpendicular line segments
If you have two parallel line segments joined together by two other parallel line segments, that's a rectangle. Triangles have to be made of line segments that are not parallel to each other.
Two parallel planes
Two planes can also be parallel, if they would never meet no matter how far they extended in all directions. A third plane, crossing both planes at right angles, would be perpendicular to the first two. The opposite sides of a cube are examples of parallel planes | 677.169 | 1 |
approximate-pi
approximate-pi
This program creates a bunch of points in a square, then draws a circle with the same radius. Then it checks which points are inside the circle. The ratio of inside to out should be around pi/4, therefor multiplying that by 4 should result in an approximation of pi. | 677.169 | 1 |
Quadrilaterals are fascinating geometric shapes that have four sides and four angles. They can take various forms, such as squares, rectangles, parallelograms, trapezoids, and rhombuses. Constructing a quadrilateral involves creating a shape with specific measurements and angles. In this article, we will explore the step-by-step process of constructing a quadrilateral, discuss different types of quadrilaterals, and provide valuable insights into their properties and applications.
Understanding Quadrilaterals
Before we delve into the construction process, let's familiarize ourselves with the different types of quadrilaterals:
Square: A square is a quadrilateral with four equal sides and four right angles.
Rectangle: A rectangle is a quadrilateral with four right angles, but its sides may have different lengths.
Parallelogram: A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.
Trapezoid: A trapezoid is a quadrilateral with one pair of parallel sides.
Rhombus: A rhombus is a quadrilateral with four equal sides, but its angles may not be right angles.
Step-by-Step Construction Process
Constructing a quadrilateral requires precision and attention to detail. Follow these steps to create a quadrilateral:
Step 1: Gather the Required Tools
Before you begin, ensure you have the necessary tools:
Compass
Straightedge (ruler)
Protractor
Pencil
Eraser
Step 2: Determine the Type of Quadrilateral
Identify the type of quadrilateral you want to construct. Each type has specific properties and requirements.
Step 3: Measure and Mark the Sides
Using the ruler, measure and mark the lengths of the sides of the quadrilateral on a blank sheet of paper. Ensure the measurements are accurate and proportional.
Step 4: Construct the First Side
Using the compass, set the desired length of the first side. Place the compass point on the starting point of the side and draw an arc that intersects the side's endpoint.
Step 5: Construct the Second Side
Set the compass to the length of the second side. Place the compass point on the endpoint of the first side and draw an arc that intersects the first arc.
Step 6: Connect the Intersecting Points
Using the ruler, draw a straight line connecting the intersecting points of the two arcs. This line represents the second side of the quadrilateral.
Step 7: Repeat Steps 4-6 for the Remaining Sides
Repeat steps 4-6 to construct the remaining sides of the quadrilateral. Ensure that the angles and lengths are accurate.
Step 8: Verify the Properties
Check if the constructed quadrilateral satisfies the properties of the desired type. For example, if you aimed to construct a square, ensure that all sides are equal in length and all angles are right angles.
Properties and Applications of Quadrilaterals
Quadrilaterals possess unique properties that make them valuable in various fields. Let's explore some of these properties and their applications:
1. Parallelograms in Architecture
Parallelograms are widely used in architecture due to their stability and symmetry. The equal and parallel sides of a parallelogram provide structural support, making it an ideal shape for buildings, bridges, and other structures.
2. Rectangles in Graphic Design
Rectangles are commonly used in graphic design due to their balanced proportions. The right angles and equal opposite sides of a rectangle create a visually pleasing and harmonious layout, making it a popular choice for posters, banners, and websites.
3. Trapezoids in Engineering
Trapezoids find applications in engineering, particularly in designing ramps and roads. The parallel sides of a trapezoid allow for smooth transitions and gradual inclines, ensuring the safety and convenience of vehicles and pedestrians.
4. Squares in Mathematics
Squares play a fundamental role in mathematics, particularly in geometry and algebra. Their equal sides and right angles simplify calculations and serve as building blocks for more complex shapes and formulas.
Q&A
Q1: Can any four-sided shape be considered a quadrilateral?
A1: No, for a shape to be classified as a quadrilateral, it must have four sides and four angles.
Q2: Are all squares rectangles?
A2: Yes, all squares are rectangles, but not all rectangles are squares. Squares have four equal sides and four right angles, while rectangles have four right angles but may have different side lengths.
Q3: Can a quadrilateral have two pairs of parallel sides?
A3: No, a quadrilateral can have at most one pair of parallel sides. If a quadrilateral has two pairs of parallel sides, it is classified as a parallelogram.
Q4: Are all rhombuses squares?
A4: No, all squares are rhombuses, but not all rhombuses are squares. Rhombuses have four equal sides, but their angles may not be right angles.
Q5: Can a quadrilateral have all sides of different lengths?
A5: Yes, a quadrilateral can have all sides of different lengths. This type of quadrilateral is called a general quadrilateral.
Summary
Constructing a quadrilateral involves a step-by-step process that requires precision and attention to detail. By following the outlined steps and understanding the properties of different types of quadrilaterals, you can create accurate and visually appealing shapes. Quadrilaterals find applications in various fields, including architecture, graphic design, engineering, and mathematics. Understanding their properties and applications can enhance your knowledge and appreciation of these geometric wonders | 677.169 | 1 |
DIRECTIONS
1. View a Construction
Click on the the checkboxes to reveal steps of the construction of each type of special quadrilateral, based on the definition specified.
2. Explore the Possibilities
The GREEN points may be moved freely.
The ORANGE points may be moved, but are constrained by the construction.
The RED points are prescribed by the construction and cannot be moved independently of the other points.
3. What Do You Notice?
For each type of special quadrilateral, what seems to always be true? That is, make some conjectures about the properties of each type of quadrilateral. WRITE down your conjectures.
SENTENCE FRAMES to get you started...
* Opposite sides are ____________.
* Opposite angles are ____________.
* The diagonals ___________.
* Every _______ is _______. | 677.169 | 1 |
TL;DR
This video explains the different types of shapes and quadrilaterals, such as circles, ovals, triangles, rectangles, squares, rhombuses, and trapezoids.
Install to Summarize YouTube Videos and Get Transcripts
Questions & Answers
Q: How can you differentiate between a circle and an oval?
Circles have a consistent length when measured from one side of the shape, across the center, to the other side. However, ovals have a shorter distance from one side to the center compared to the distance from the other side to the center.
Q: What is a quadrilateral?
A quadrilateral is a shape with four sides. It encompasses various types of shapes, including rectangles, squares, rhombuses, and trapezoids.
Q: How can you identify a rectangle?
A rectangle has four right angles or square corners. Each side can have different lengths, but the corners must be 90 degrees.
Q: What is the difference between a rhombus and a rectangle?
A rhombus has four sides of the same length, but its corners are not right angles. In contrast, a rectangle has four right angles but may have sides of different lengths.
Summary & Key Takeaways
The video discusses the characteristics and definitions of various shapes, including circles, ovals, triangles, and quadrilaterals.
It explains that circles have a constant length when measured between two points through the center, while ovals do not.
The video also introduces the concept of quadrilaterals, which are shapes with four sides, and explores different types such as rectangles, squares, rhombuses, and trapezoids. | 677.169 | 1 |
The first three books of Euclid's Elements of geometry, with theorems and problems, by T. Tate
Inni boken
Resultat 1-5 av 65
Side 6 ... BC is equal to BA : But it has been proved that CA is equal to AB ; therefore CA , CB are each of them equal to AB ; but things which are equal to the same are equal to one another ( Axiom 1. ) ; therefore CA is equal to wherefore CA ...
Side 7 ... equal to a given straight line . Let A be the given point , and BC the given straight line ; it is required to draw from the point A a straight line equal to BC . B K Ꭰ From the point A to B draw ( Post . 1. ) the straight line AB ...
Side 8 ... equal to c , the less . From the point a draw ( 1. 2. ) the straight line AD equal to C ; and E B from the centre A ... BC shall be equal to the base EF ; B and the triangle ABC to the A.A CE T triangle DEF ; and the other angles , to ...
Side 9 ... equal to DF ( Hyp . ) . But the point в coincides with the point E ; wherefore the base BC shall co- incide with the base EF , because the point в coin- ciding with E , and c with F , if the base BC does not coincide with the base EF ...
Side 10 ... equal to a C , and let the straight lines A B , AC be produced to D and E , the angle ABC shall be equal to the ... BC is common to the two angles BFC , CGB ; where- fore the triangles are equal ( 1. 4. ) , and their re- maining angles ... | 677.169 | 1 |
ATAN
In this article
Returns the arctangent, or inverse tangent, of a number. The arctangent is the angle whose tangent is number. The returned angle is given in radians in the range -pi/2 to pi/2.
Syntax
ATAN(number)
Parameters
Term
Definition
number
The tangent of the angle you want.
Return value
Returns the inverse hyperbolic tangent of a number.
Remarks
To express the arctangent in degrees, multiply the result by 180/PI( ) or use the DEGREES function.
Example
Formula
Description
Result
= ATAN(1)
Arctangent of 1 in radians, pi/4
0.785398163
= ATAN(1)*180/PI()
Arctangent of 1 in degrees
45 | 677.169 | 1 |
Statistics
Notes
Congruence of Circles:
Observe the circles in the figures above. Draw similar circles of radii 1 cm, 2 cm, 1 cm, and 1.3 cm on a paper and cut out these circular discs. Place them one upon the other to find out which ones coincide exactly.
The circles in (a) and (c) coincide.
Circles in fig (b) and (c) and in fig (a) and (d) do not coincide.
Circles which coincide exactly are said to be congruent circles.
Circles of equal radii are congruent circles.
If you would like to contribute notes or other learning material, please submit them using the button below. | 677.169 | 1 |
The Many Shapes of Geometry
Most people think of geometry as squares, circles, and triangles. And while it's true that those are some of the most basic shapes in geometry, there's a lot more to it than that. In fact, there are literally hundreds of different shapes that fall under the umbrella of geometry. In this blog post, we'll take a look at some of the most popular ones.
Circles
We'll start with circles since they're probably the most well-known geometric shape. A circle is defined as a closed curve formed by a set of points equidistant from a central point. Circles are often used in mathematical formulas and equations.
Triangles
Triangles are another very popular geometric shape. A triangle is defined as a three-sided polygon. There are many different types of triangles, including equilateral, isosceles, and scalene triangles.
Squares and Rectangles
Squares and rectangles are both four-sided polygons. A square is a rectangle with all sides of equal length, while a rectangle has two sides of equal length (the other two sides are not necessarily the same length).
Other Popular Geometry Shapes
Other popular geometry shapes include pentagons (five-sided polygons), hexagons (six-sided polygons), octagons (eight-sided polygons), and decagons (ten-sided polygons). These shapes are often used in architecture and construction.
There you have it! Just a sampling of the many different shapes that fall under the geometry umbrella. As you can see, there's a lot more to geometry than just squares, circles, and triangles. So if you're ever feeling stumped by a math problem that involves one of these shapes, don't hesitate to ask your teacher for help.
FAQ
What are the 8 types of geometry?
The eight types of geometry are: Euclidean, Analytic, Transformational, Algebraic, Discrete, Combinatorial, Differential, and IntuitiveDiscrete geometry is the study of geometry in a discrete setting, such as on a computer or a graph. It has applications to computer science and networking.
Combinatorial geometry is the study of geometric objects such as points, lines, and planes using combinatorial methods. It has applications to data mining and machine learning.
Differential geometry is the study of geometry using calculus. It is used in physics and engineering.
Intuitive geometry is the study of geometry using intuition and visual aids. It is often used in education and for art and design.
What are 7 basic geometric forms?
The seven basic geometric forms are: point, line, plane, circle, sphere, cylinder, and cone. Each formWhat are the 4 types of geometry?
The four types of geometry are: Euclidean, Analytic, Transformational, and AlgebraicWhat are the 10 basic geometry terms?
The ten basic geometry terms are: point, line, plane, circle, sphere, cylinder, cone, Euclidean, Analytic, and Transformational. Each term | 677.169 | 1 |
Welcome to the isosceles triangle angles calculator, where we'll show you how to calculate the angles of an isosceles triangle. Along the way, we hope to teach you more about:
What kinds of angles an isosceles triangle has;
What the vertex angle of an isosceles triangle is; and
What angles an isosceles right triangle has.
How do I use the isosceles triangle angles calculator?
The isosceles triangle angles calculator gets to the point without cutting any triangles' corners. Here's how to use it:
Enter the length of your triangle's legs and the base length (aaa and bbb, respectively). If you need to enter the value in a different unit, click on the unit to change it, then enter the length.
See how the calculator instantly works out the vertex angle and base angles (β\betaβ and α\alphaα). If you desire the result in a different unit, you can change it by clicking on the unit.
For example, consider an isosceles triangle with 4 cm legs and a 5 cm base. Enter the legs and base length in our calculator. Right away, it tells you that the vertex angle β=77.4°\beta = 77.4\degreeβ=77.4°, and the base angle α=51.3°\alpha = 51.3\degreeα=51.3°.
The calculator can work backward, too! Try inputting your mystery triangle's angles with one side's length (the isosceles triangle angles calculator needs some sense of proportion) and see how it works out the remaining side's length.
Say you want to calculate the legs of a right isosceles triangle whose hypotenuse (base) is 12 feet long. Here, you enter the vertex angle β=90°\beta = 90\degreeβ=90°. Next, click on the base's unit and change it to feet. Enter the base length b=12ftb = 12\rm\ ftb=12ft. Instantly, you can learn that the legs are 8.49 ft each.
If you still want to learn what an isosceles triangle is and how to calculate its angles, read on!
What is an isosceles triangle? What kind of angles does an isosceles triangle have?
An isosceles triangle is a triangle with two sides of equal length. We usually call these two sides the "legs" (aaa below) and the remaining side the "base" (bbb).
An isosceles triangle. See how the legs marked aaa are the same length, and the corners marked α\alphaα are the same angle?
Because the legs of an isosceles triangle are the same length, the two angles they form with the base are also equal. We call the angles adjacent to the base side the base angles (α\alphaα) and the remaining angle the vertex angle (β\betaβ).
How do I find the angles of an isosceles triangle?
Finding the anglesα\alphaαandβ\betaβof an isosceles triangle is easy — all you need is some geometry tricks!
Findaaaandbbb, the length of the isosceles triangle's base and legs.
Split the isosceles triangle down its axis of symmetry (i.e. from its vertex angle straight down the middle of the base) to obtain two mirrored right triangles.
Use trigonometry to work out the angles, which will be 90∘90^\circ90∘, α\alphaα, and β/2\beta/2β/2.
What is an isosceles right triangle?
An isosceles right triangle is the child of a right triangle and an isosceles triangle, and so it has all of its parents' attributes. It has two equal sides and one angle (in this case, the vertex angle) that is 90∘90^\circ90∘. The two base angles are then 45∘45^\circ45∘ each.
Related calculators
If you found this isosceles triangle angles calculator handy, you may also like:
Meet the creator of isosceles triangle angles calculator
I'm Rijk, a data scientist and an expert at creating scientific tools. Through rigorous user data analysis, I found people had difficulty calculating isosceles triangle angles and immediately created a solution. Ever since, this calculator has helped numerous people, and I hope you had fun using it, too!
We put extra care into the quality of our content so that they are as accurate and reliable as possible. Each tool is peer-reviewed by a trained expert and then proofread by a native speaker. You can learn more about our standards on our Editorial Policies page.
FAQ
What is the vertex angle of an isosceles triangle?
The vertex angle of an isosceles triangle is the angle formed by the triangle's two legs (the two sides that are of equal length). It is unique in the triangle unless all three sides are equal and the triangle is equilateral.
Can an isosceles triangle have a 90 degree angle?
Yes — an isosceles triangle can have a 90-degree angle. This angle would be its vertex angle. We can call an isosceles triangle with a 90-degree angle a right isosceles triangle. The base angles of an isosceles triangle can't be 90 degrees — a shape with two or more 90° angles can't even be a triangle at all!
What are the angles of an isosceles triangle with a vertex angle of 90°?
If an isosceles triangle has a vertex angle β = 90°, we only need to calculate one more angle — the base angle, α, which features twice.
The sum of a triangle's angles is 180°, i.e.: 2α + β = 180°.
Make α the subject of the equation: α = (180° − β) / 2
Substitute β = 90°: α = (180° − 90°) / 2
Work out the equation to obtain α = 45°.
Rijk de Wet
Leg (a)
in
Base (b)
in
Vertex angle (β)
deg
Base angle (α)
deg
Check out 19 similar triangle calculators 🔺
30 60 90 triangle45 45 90 triangleArea of a right triangle… 16 more
People also viewed…
Isosceles triangle area
The isosceles triangle area calculator will calculate the area of an isosceles triangle based on the lengths of its sides and its height. | 677.169 | 1 |
Construction of Pie Chart
Now we will discuss about the construction of pie chart or pie graph. In
brief let us recall about, what is a pie chart?
The pie chart is a pictorial representation of data relative to a whole. Each portion in the circle represent an element of the collected data. The pie chart represents the composition of various elements in a whole. The total value of the pie chart is always 100%. Each portion in the circle shows a fraction or percentage of the total.
Pie chart is a circular graph which is used to represent data. In this :
● Various observations of the data are represented by the sectors of the circle.
● The total angle formed at the centre is 360°.
● The whole circle represents the sum of the values of all the components.
● The angle at the centre corresponding to the particular observation component is given by
●
If the values of observation/components are expressed
in percentage,
then the centre angle corresponding to particular
observation/component is given by
How to construct a pie chart?
Steps
of construction
of pie chart for a given data:
●
Find the central angle for each component using the
formula given on
the previous page.
● Draw a circle of any radius.
● Draw a horizontal radius
● Starting with the horizontal radius, draw radii, making
central angles
corresponding to the values of respective components.
● Repeat the process for all the components of the given
data.
● These radii divide the whole circle into various
sectors.
● Now, shade the sectors with different colors to denote
various
components.
● Thus, we obtain the required pie chart.
Here the pie chart shows the type of books preferred by the students of a class.
Given below is a pie chart showing the type of books preferred by the students of a class. Observe the chart and answer the questions given below.
(i) What fraction of students like Comic books?
Answer: The sector in the pie chart shows = \(\frac{1}{4}\)
(ii) What fraction of students like Story books?
Answer: The sector in the pie chart shows = \(\frac{1}{2}\)
(iii) Which 2 books have equal fraction of preference?
Answer: Poem and Puzzle books.
(iv) What fraction of students does not like story books?
Answer: \(\frac{1}{2}\)
(iv) Which type of book do you like to read?
Answer: Comic books
Solved Example on Construction of Pie Chart/Pie Graph:
1. The following table shows the numbers of hours spent by a child on different events on a working day.
Represent the adjoining information on a pie chart
Activity
No. of Hours
School
6
Sleep
8
Playing
2
Study
4
T. V.
1
Others
3
The central angles for various observations can be calculated as:
Activity
No. of Hours
Measure of central angle
School
6
(6/24 × 360)° = 90°
Sleep
8
(8/24 × 360)° = 120°
Playing
2
(2/24 × 360)° = 30°
Study
4
(4/24 × 360)° = 60°
T. V.
1
(1/24 × 360)° = 15°
Others
3
(3/24 × 360)° = 45°
Now, we shall represent these angles within
the circle as different sectors. Then we now make the pie chart:
2. The favourite flavours of ice-cream
for the children in a locality are given in percentage as follow. Draw the pie
chart to represent the given information
Flavours
% of Students Prefer the Flavours
Vanilla
25 %
Strawberry
15 %
Chocolate
10 %
Kesar-Pista
30 %
Mango Zap
20 %
The central angles for various observations can be calculated as:
Flavours
% of Students Prefer the Flavours
Measure of Central Angles
Vanilla
25 %
(25/100 × 360)° = 90°
Strawberry
15 %
(15/100 × 360)° = 54°
Chocolate
10 %
(10/100 × 360)° = 36°
Kesar-Pista
30 %
(30/100 × 360)° = 108°
Mango Zap
20 %
(20/100 × 360)° = 72°
Now,
we shall represent these angles within a circle to obtain the required pie
graph.
Questions and Answers on Construction of Pie Chart:
1. 80 girls were asked about their favourite sport? The collected information is given in the table. Read the information and label the graph.
Sports
Number of Girls
Badminton
41
Tennis
9
Swimming
23
Volleyball
7
Answer:
2. 100 students of a school were surveyed for their favourite activity. The information is represented in a pie chart as given below. Observe the chart and answer the questions | 677.169 | 1 |
The sine of 245 degrees is -0.906.
Sure, here's a brief introduction for your blog post:
Exploring the Sine Function: What is the Sine of 245 Degrees?
In the world of Mathematics education, understanding trigonometric functions is essential. In this article, we delve into the concept of the sine function and specifically explore the calculation of the sine of 245 degrees. By breaking down the process step by step, we aim to provide clarity and insight into this fundamental aspect of trigonometry. Join us as we unravel the mysteries of the sine function and its application in real-world scenarios.
Understanding Trigonometric Functions
Trigonometric functions are essential tools in understanding the relationships between angles and sides of a right triangle. They provide a way to calculate unknown sides or angles based on known information, and they also have applications in fields such as engineering, physics, and navigation.
The Sine Function and Angles
The sine function is one of the primary trigonometric functions that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. When dealing with angles greater than 180 degrees, it's important to understand how to interpret and calculate the sine function for these angles.
Calculating the Sine of 245 Degrees
When finding the sine of 245 degrees, it's crucial to understand the concept of reference angles and how they relate to the sine function for angles beyond 180 degrees. By using the properties of the unit circle and periodicity of trigonometric functions, we can accurately determine the sine of 245 degrees.
Real-world Applications of Trigonometric Functions
Understanding trigonometric functions, including the sine function, is not only important in mathematics education but also in real-world scenarios. From analyzing periodic phenomena to calculating distances and angles in various fields, a solid grasp of trigonometric functions is invaluable.
frequently asked questions
How do you calculate the sine of 245 degrees in Mathematics education?
You can calculate the sine of 245 degrees by using the formula sin(θ) = sin(θ - 360°) to find the equivalent angle between 0 and 360 degrees, then evaluate the sine function for that angle.
What are the steps to find the sine of 245 degrees in a Mathematics education setting?
The steps to find the sine of 245 degrees in a Mathematics education setting are to convert 245 degrees to its equivalent angle in the first quadrant (65 degrees) and then use the trigonometric function sin(θ) to find the sine value, which is 0.9063.
Why is understanding the sine function important in Mathematics education when dealing with angles like 245 degrees?
Understanding the sine function is important in Mathematics education when dealing with angles like 245 degrees because it allows students to calculate the vertical component of the angle's position, which is crucial for various mathematical and real-world applications.
How can a Mathematics educator explain the concept of finding the sine of 245 degrees to students?
A Mathematics educator can explain the concept of finding the sine of 245 degrees to students by using the unit circle and identifying the reference angle of 65 degrees. Then, they can show how to use the co-function identity to find the sine of 245 degrees.
Are there any common misconceptions or challenges that students face when learning about the sine of 245 degrees in Mathematics education?
One common misconception is that the sine of 245 degrees is negative. This misconception arises from misunderstanding the quadrant in which 245 degrees lies. In reality, the sine of 245 degrees is positive, as it falls within the second quadrant.
In conclusion, understanding the concept of sine and its relationship to angles is crucial in Mathematics education. The sine of 245 degrees can be calculated using the unit circle or reference angles, providing a deeper insight into trigonometric functions. Emphasizing the importance of this knowledge can enhance students' understanding of trigonometry and its applications in various | 677.169 | 1 |
Online Pythagorean Theorem Calculator for Geometry Enthusiasts
Humans have been working on geometry for years. Geometry describes the shape of the objects. This theorem can be applied to multiple ideas. It also explains the structure of the right-angle triangle. An onlinePythagorean Theorem Calculator can be a very helpful tool for anyone who has to perform rapid calculations, whether they are professionals or students learning geometry. We will examine the Pythagorean Theorem, its importance in geometry, and how internet calculators make it easier for geometry fans to use in this complete introduction.
Acquiring knowledge of the Pythagorean Theorem
According to the Pythagorean Theorem, the square of the length of the hypotenuse in a right triangle equals the sum of the squares of the lengths of the other two sides. The hypotenuse is the side that faces the right angle. It has the following mathematical expression:
a² + b² = c²
Where:
The measurements of the two shorter sides also referred to as the right triangle's legs, are a and b.
The hypotenuse's length is given by c.
This theorem has significant effects on geometry, trigonometry, and several practical applications. We can use it to measure distances, establish angles, and address a variety of geometric issues.
The Pythagorean Theorem's Importance
It can be applied in multiple domains.
Construction and Architecture
The theory is used by architects and contractors to guarantee the stability and structural integrity of buildings. It aids in measuring diagonal distances and confirming the squareness and levelness of constructions.
Orientation
People are using it for navigation purposes.
Technical
Engineers use the theorem in a variety of fields, including electrical engineering (to calculate circuit voltages) and civil engineering (for constructing bridges and roads).
Surveying and Mapping
The theorem is used by surveyors and cartographers to produce precise maps, determine land areas, and determine the separations between locations on the surface of the Earth.
Health
The Pythagorean Theorem is used in medical imaging, such as CT scans and MRI, to compute distances inside the human body and establish the sizes of organs or tumors.
Physics
The theorem is used by physicists in several computations involving forces, motion, and mechanics.
The Pythagorean Theorem in Use
You must be aware of the lengths of the first two sides of a right triangle to use the Pythagorean Theorem. You then seek to determine the length of the third side. Let's divide the procedure into stages:
Make sure you are working with a right triangle, which has one angle that is 90 degrees, by identifying it.
Give the triangle's sides names by labeling them. The hypotenuse, commonly denoted by the letter "c," is the side that is opposite the right angle. The "a" and "b"-designated legs are the other two sides.
Describe your Knowledge
Decide which sides (a and b) you are aware of the lengths of and which side (c) you are trying to find. Apply the theorem: Determine the length of the missing side using the equation a2 + b2 = c2. The function of online calculators for Pythagorean theorem
While manually computing the Pythagorean Theorem is a useful ability, it may also be time-consuming and error-prone. Online Pythagorean Theorem calculators make the process easier and deliver precise results quickly. Here is how they function:
The lengths of the two known sides ('a' and 'b') can normally be entered in the fields found on online calculators.
Calculate
Once the values have been entered, you press the "calculate" button, and the calculator immediately determines the length of the third side (designated by the letter "c") by applying the Pythagorean Theorem formula.
Results Display
The calculated outcome is shown on the screen, frequently with the option to round the result to a particular number of decimal places for accuracy.
Clear and Reset
The majority of online calculators let you start over for new calculations by erasing the input fields.
Flexibility
Using a Pythagorean Theorem Calculator Online
It is simple to use an online Pythagorean Theorem calculator. The steps are as follows:
How to use the calculator Use your chosen search engine to look up "Pythagorean Theorem Calculator" or go to a website that deals with maths to find one there.
Input recognized values: Fill up the appropriate fields on the calculator with the lengths of the two known sides ('a' and 'b').
Calculate
To determine the length of the third side ('c'), click the "calculate" or "solve" button. View outcomes The screen will show the calculated length of the omitted side ('c'). Results should be rounded to a certain number of decimal places if necessary for accuracy.
Clear and Reset
Use the "clear" or "reset" button to clear the input fields if you need to make more computations.
Input recognized values: Fill up the appropriate fields on the calculator with the lengths of the two known sides ('a' and 'b').
Professional Sectors
These calculators are used in the work of engineers, architects, surveyors, and designers to tackle challenging geometrical issues.
Homemade Projects
People use these calculators to organize and carry out projects that require accurate measurements, from DIY enthusiasts to crafters.
Conclusion
A fundamental idea in geometry known as the Pythagorean Theorem has a wide range of uses, from architecture to medicine. Online Pythagorean Theorem calculators make it easier to apply this theorem and get prompt, precise results. | 677.169 | 1 |
Given circle c with center A and a chord HI, and a point G within segment HI, construct circle d with center C such that the radical axis passes through G and point F lies on both d, line HI and line CA.
TL;DR Given the things in blue, construct the things in red.
Intuition tells me that there should be an unique solution given the level of constraint. I have also constructed one point J on the red circle.
A purely geometric construction and proof, not utilizing analytic geometry, would be appreciated.
1 Answer
1
Once you constructed point $J$, you can construct $F$ as the intersection of the line $HI$ and the line perpendicular to $AJ$ passing through $J$. This is because $AF$ is a diameter of $d$. The rest of the construction is easy and I leave it to you. | 677.169 | 1 |
Basic Constructions
Are you lost when it comes to using a compass? A geometry compass, that is! They can be used to draw basic math constructions, and even artwork! Ms. Mars shows you step-by-step steps to creative fun!
categories
Congruence, Transformations, Similarity, Plane Geometry (2D)
subject
Math
learning style
Kinesthetic, Visual
personality style
Otter, Beaver
Grade Level
High School (9-12)
Lesson Type
Dig Deeper
Lesson Plan - Get It!
Audio:
Compass Artwork
Who said math wasn't a creative subject?
Let's disprove that notion.
Download the Compass Artwork document from Downloadable Resources in the right-hand sidebar. By the time you finish this lesson, your creative juices will be flowing!
Take a moment to look at the document.
Don't do anything yet, just look.
Does anything come to mind?
Do you think this is all strictly technical?
Now, it's time to get started!
Below is a list of vocabulary words that you need to be familiar with for this lesson.
Construction is the act of using a compass and straightedge to make a geometric figure.
Compass is a geometric tool used to draw circles and arcs.
Straightedge is a ruler without markings.
Arc is a connected section of a circumference of a circle.
The purpose of this lesson is to continue to add to your geometry vocabulary, help you become familiar with how to use a compass and a straightedge, and teach you how to draw the four basic constructions.
Before you begin drawing any of the constructions, it is important for you to understand the different parts of a compass and how to use it.
Above is a picture of a geometry compass.
Notice that this geometric tool has a point on the end of one leg, a pencil inserted into the other leg, and a knob where the pencil is inserted.
Also, notice that the compass has a black handle sticking out from the top. The two legs of a compass have the ability to spread farther apart or come closer together.
Use your own compass to see how easy it is to make the legs farther apart, then bring them closer together (Note: the point on the one leg can hurt, so avoid touching it.).
When you are getting ready to use your compass to draw a construction, you should go through a couple of steps:
Make sure the pencil has a point on the end so it will make a clean mark.
If your pencil needs to be sharpened, simply turn the knob on the pencil leg and loosen the pencil until it comes out easily. Then, sharpen the pencil, replace it into the holder, and tighten the knob.
Prior to using your compass, make sure the point of your pencil and the point on the other leg are equal.
The following information contains the four basic constructions — and short activities with each — so you can try working with your compass. Before starting the activity, discuss with your teacher the definitions of segment and ray.
Congruent Segments
This basic construction activity will help you learn how to draw two congruent segments:
Congruent Angles
This basic construction will teach you how to draw two congruent angles:
Perpendicular Bisector
This basic construction will teach you how to draw a perpendicular bisector for a segment:
Angle Bisector
This last construction will show you how to draw an angle bisector and split the angle into two congruent angles:
The following Got It? section offers activities for you to practice your new skills. | 677.169 | 1 |
Rotation about a Point
Rotation is a Rigid-Motion Transformation
about a Point of Rotation at a given Angle of Rotation.
Triangle ABC is rotated about Point O (0,0)
Move Slider α for an Angle of Rotation between 0° and 360°.
1. What kind of a triangle is ABC ? _____________________________________.
2. What are the Coordinates for Points A, B, and C?
Point A ( _____ , _____ ), Point B ( _____ , _____ ), and Point C ( _____ , _____ ).
3. Move the slider to 90°, what are the Coordinates for Points A', B', and C'?
Point A' ( _____ , _____ ), Point B' ( _____ , _____ ), and Point C' ( _____ , _____ ).
4. Move the slider to 180°, what are the Coordinates for Points A', B', and C'?
Point A' ( _____ , _____ ), Point B' ( _____ , _____ ), and Point C' ( _____ , _____ ).
5. Move the slider to 270°, what are the Coordinates for Points A', B', and C'?
Point A' ( _____ , _____ ), Point B' ( _____ , _____ ), and Point C' ( _____ , _____ ).
6. Compare and contrast coordinates for the above rotations:
What remained the same _______________________________________________________
What changed? _______________________________________________________
7. What do you notice about the slopes of the side AB and side BC ?
____________________________________________________________________________________
8. Move Point A to change the orientation of the triangle ABC.
Is the above relationship between coordinate points the same?
What is Different ? ______________________________________________________________
____________________________________________________________________________________
Next, be sure pre-image and rotated image is not onto itself (rotate 90°).
Place cursor on A' of rotated image, touch trackpad with two fingers for additional options, and select [Trace on],
repeat for points B' and C'.
Place cursor on slider α, touch trackpad with two fingers for additional options, and select [Animation On] .
What relationship exist between the pre-image points and the rotated image points with respect to the origin ?
__________________________________________________________________________________________
To reset Trace, deselect [Animation On] and use two fingers on trackpad to scroll in or out.
Move the Point of Rotation away from the origin.
Does the same relationship exist between the pre-image points and the rotated image points with respect to the origin ?
__________________________________________________________________________________________ | 677.169 | 1 |
We need to decide whether to use the \( + \) or the \(-\) in front of the formula in \( (I) \). In part a) we found the sign of \( \sin \left(\dfrac{\theta}{2} \right) \) to be positive, hence
\( \qquad \sin \left(\dfrac{\theta}{2} \right) = \color{red}{ + } \sqrt {\dfrac{1 - \cos \theta} {2}} \)
We are given \( \tan(\theta) \) and the quadrant of \( \theta \).
Use the definition of \( \tan(\theta) \) in a right triangle
\( \qquad \tan(\theta) = \dfrac{\text{Opposite Side}}{\text{Adjacent Side}} \)
Solutions to the Above Exercises
We need to find the quadrant of \( \dfrac{\theta}{2} \) to select the + or - in the formula and we also need to determine \( \cos \theta \).
\( \theta \) is in quadrant 3 (given) may be written using an inequality as
\( \qquad \pi \lt \theta \lt \dfrac{3 \pi}{2} \) | 677.169 | 1 |
A Supplement to the Elements of Euclid
ﭼﻥﻕﮔﻐﮪﻝﮩﻝ ﮩﮪﺅ ﻗﻠﻗﻣﻑﺅ
ﭼﻭﺅﮪﮒﻣﻏﮩﮞﻕﮪﻕ 6 - 10 ﻕﻭﮰ ﮪﻕ 29.
ﺽﮒﻣﻑﻛﻕ 33 ... side EB ; and join A , D : Then ( S. 7. 1. ) CA falls within EB ; the CAE is a right , the than a right ; .. ( E. 17 ... number of triangles have a right angle for their common vertical angle , and have equal hypotenuses , the locus of ...
ﺽﮒﻣﻑﻛﻕ 36 ... figure ADBC are right ; and it has been proved that all its sides are equal ; .. ( E. 30. def . 1. ) ADBC is a square . PROP . XXXI . 44. THEOREM . If either of the acute angles of a given right - angled triangle be divided into any number ...
ﺽﮒﻣﻑﻛﻕ 56 ... figure EFGH is equilateral ... number of equal parts , the straight lines , joining the oppo- site points of division , shall also divide the dia- meter of the parallelogram into the same number of equal parts . Let the two opposite sides ...
ﺽﮒﻣﻑﻛﻕ 68 ... number of sides being given , to find an equal rec- tilineal figure , which shall have the number of its sides less , or greater , by one , than that of the given figure . First , let ABCDE be a given rectilineal figure : A B E C D F It ...
ﺽﮒﻣﻑﻛﻕ 366 - If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the...
ﺽﮒﻣﻑﻛﻕ 92ﺽﮒﻣﻑﻛﻕ 367ﺽﮒﻣﻑﻛﻕ 104 | 677.169 | 1 |
Enjoy this nicely organized worksheet that puts together multiple problems regarding trapezoid proofs. The resource can be used as a guide that begins with proving properties and ends with solving for measures of line segments.
Stuck with triangle proofs? Take a 180° and provide learners with a guided worksheet that tests their knowledge with triangle inequalities. The questions require different types of proofs that range in levels of difficulty.
Stuck trying to remember the formal language of a geometric proof? Never fear, this handout has them all ready to go. The reasons are sectioned by topic so this handy guide is ready when you are to tackle those two column proofs.
This amazingly extensive unit covers a wealth of geometric ground, ranging from constructions to angle properties, triangle theorems, rigid transformations, and fundamentals of formal proofs. Each of the almost-forty lessons is broken...
Learners watch a quick review on the Parallel and Perpendicular Lines unit. The pupils work through ten review problems ranging from identifying types of angles formed by parallel lines and transversals to writing equations of parallel...
Show it can all be proved. Scholars learn the converses of the properties of parallel lines. Using the converses, pupils determine which lines are parallel based on angle measurements and practice using a flow proof to show that two...
Work within a parallel universe. Scholars learn about the relationships of angles when two parallel lines are intersected by a transversal. They see how to find the measurements of all eight angles by knowing the measure of one angle....
The proof is in the review. Individuals watch a short review of the content from the unit to prepare for the unit exam. The review covers inductive reasoning, conditional and related statements, and two-column algebraic and geometric...
Proofs may be as easy as 1, 2, 3 ... maybe. Pupils participate in creating four example proofs. The presentation uses a list of geometric properties to develop the proofs by filling in both the statements and reasons. Scholars practice...
Add a little algebra to the geometry. Class members learn about the Addition Postulate for segments and angles. The pupils use their knowledge of solving equations to find lengths of segments and measures of angles. Individuals apply...
Complement the class by identifying pairs of angles. With a vocabulary lesson, pupils learn the names and definitions of special angle pairs including adjacent, vertical, complementary, supplementary, and linear pairs. Using the...
Transform the classroom into a coordinate plane. Scholars walk along parallel lines drawn on a coordinate plane on the classroom floor. They measure angles and slopes and use the results to develop criteria that would make the lines...
Perhaps planning a city isn't so difficult after all. Scholars first perform geometric constructions and investigate how parallel lines are useful in real-world situations. They then work on a city design project, drawing street maps,...
Use the parallel lines to find your way. After first reviewing geometric constructions and the relationships between angles formed by parallel lines and a transversal, young mathematicians write proofs for theorems relating to parallel...
There's no parallel to a great resource. Scholars take part in three lessons in which they investigate parallel lines in real-world situations. They search for and take pictures of parallel lines in and around the school, develop artwork...
It's just not enough to know that something is true. Part of a MVP Geometry unit teaches young mathematicians how to write flow proofs and two-column proofs for conjectures involving lines, angles, and triangles.
Bank on geometry to line up the shot. The resource asks the class to determine the location to bank a cue ball in a game of billiards. Using their knowledge, class members determine where to hit the bumper to make a shot and discuss...
Transform proofs using triangle rotations. By rotating a triangle around various points, class members develop proofs. Participants prove relationships of alternative interior angles formed by parallel lines and the sum of the interior...
Navigate your way through a lesson on types of lines. Individuals drag line segments to illustrate paths between pairs of houses on an interactive map. They determine if these line segment pairs are intersecting or
parallel.
If lines aren't parallel or perpendicular, then what are they? An interactive lets users rotate a line to change its orientation
with respect to another line. It then indicates whether the
lines are parallel, perpendicular, or...
Geometric figures are perfect to use for proofs. Scholars prove conjectures about whether given points lie on a triangle and about midpoints. They use a provided dialogue among fictional students to frame their responses.
Explore angle relationships associated with transversals. Pupils construct parallel lines with a transversal and find the measures of the angles formed. They figure out how the different angles are related before constructing... | 677.169 | 1 |
Teaching Kids About Acute Angles
The acute angle is small, less than 90˚ (degrees). It is one of many primary types of angles, along with a right angle (equal to 90˚), an obtuse angle (between 90˚ and 180˚), and a reflex angle, which measures more than 180˚ but less than 360˚.
You can remember the definition of an acute angle by thinking of this mnemonic: an acute angle is small.
What is an angle?
Firstly, what is an angle? We have to understand angles before we can start thinking about acute angles.
In geometry, an angle can be defined as the figure formed by two rays meeting at a common endpoint. The two rays — lines — that meet at the endpoint are called the 'arms of the angle.' An angle measures a turn, calculated in degrees or °. There are 360° in a complete turn. The endpoint is also known as the vertex.
You find out the size of an angle using a protractor, which is a bit like a ruler. It is a semicircular piece of flat, transparent plastic with lines marked on it. Then, you can line up the lines with an angle to measure the degrees of the angle.
What are the five types of angles?
A few different types of angles vary in size and measurement. Therefore, children need to learn about the other angles.
Acute angles are angles of measurement less than 90˚.
Obtuse angles are greater than 90° but less than 180°.
Reflex angles are greater than 180°.
Right angles are exactly 90˚ in measurement.
Straight angles are 180° exactly, so they are a straight line.
Complete angles are a whole angle, exactly 360˚ in measurement, and so are a circle | 677.169 | 1 |
Translation, rotation, and reflection are ___________________.
Isometry
Non-isometry
Symmetry
Asymmetry
Hint:
Translation, rotation and reflection are all terms used to describe the transformation of shapes in maths. This means the movement of a shape around a fixed point or across a mirror line. The shape remains the same, but is translated, rotated or reflected.
The correct answer is: Isometry
Translation, rotation, and reflection are isometry | 677.169 | 1 |
ত্রিভুজবিদ্যা
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. | 677.169 | 1 |
articlescribble
Whats the distance from point y to qx in the figure
Accepted Solution
A:
The distance from point [tex]y[/tex] to [tex]WX[/tex] is just the measure of the line segment [tex]YZ[/tex]. Notice that [tex]YZ[/tex] is one of the sides of the right triangle [tex]XYZ[/tex], so to find its length we are going to use the Pythagorean theorem: [tex](YZ)^2=(XY)^2-(XZ)^2[/tex] [tex](YZ)^2=(10 \sqrt{2} )^2-10^2[/tex] [tex](YZ)^2=200-100[/tex] [tex](YZ)^2=100[/tex] [tex]YZ= \sqrt{100} [/tex] [tex]YZ=10[/tex] | 677.169 | 1 |
Mensuration of lines, surfaces, and volumes
From inside the book
Results 1-5 of 24
Page 6 ... measure of an angle at the centre of a circle is the same as the numerical measure of its intercepted arc , if the adopted unit of angle is the angle at the centre which intercepts the adopted unit of arc ...... XIV . To find the number ...
Page 9 ... measure a surface , is to find how often it contains the unit of surface ; similarly , to measure a volume , is to find how often it contains the unit of volume . In measuring such magnitudes , however , we do not do so by directly ...
Page 27 ... MEASURE OF ANGLES . PROP . XIII . - The numerical measure of an angle at the centre of a circle is the same as the numerical measure of its intercepted arc , if the adopted unit of angle is the angle at the centre which intercepts the ...
Page 28 ... measured by ' is used for has the same numerical measure as . ' Cor . 1. - An angle , ABC , at the circumference of a circle is measured by half the arc AC on which it stands , since it is equal to half the angle AOC at the centre ...
Page 29 ... measured by referring it to the unit angle , or , which is the same thing , when its arc is measured by referring it to the unit arc , the measure we obtain is said to be the circular measure of the angle . There is another method | 677.169 | 1 |
In this blog post, we will be discussing about Screenshot math solver. The three main branches of trigonometry are Plane Trigonometry, Spherical Trigonometry, and Hyperbolic Trigonometry. Plane Trigonometry is concerned with angles and sides in two dimensions, while Spherical Trigonometry deals with angles and sides on the surface of a sphere.Screenshot math solver Hyperbolic Trigonometry is concerned with angles and sides in three dimensions. The applications of trigonometry are endless, making it a vital tool for anyone who wants to pursue a career in mathematics or science. | 677.169 | 1 |
Can a right angle also be obtuse?
A right triangle cannot be obtuse because of the sizes of the angles therein.
Why can't a right triangle have an obtuse angle?
When an angle of a triangle is 90 degrees, the triangle cannot have an obtuse angle. The other two must each be less than 90 degrees (90 deg + 89 deg + 1 deg = 180 deg). It therefore follows that they must both be less than 90 degrees and so must both be acute.
What makes an angle right acute or obtuse?
Acute Angle – An angle less than 90 degrees. Right Angle – An angle that is exactly 90 degrees. Obtuse Angle – An angle more than 90 degrees and less than 180 degrees. Straight Angle – An angle that is exactly 180 degrees.
What is the difference between right acute and obtuse triangles?
A right triangle has one angle that's 90° and a corner that looks like an L. Obtuse triangles have one angle that's greater than 90°. In acute triangles, all the angles are less than 90°.
Can an obtuse triangle and a right triangle be similar?
A triangle cannot be right-angled and obtuse angled at the same time. Since a right-angled triangle has one right angle, the other two angles are acute.
Does two right angles make an obtuse angle?
An obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees.
How many degrees is an obtuse angle?
An obtuse angle refers to an angle that is more than 90 degrees but less than 180 degrees. These angles extend past a right angle. Learn more about obtuse angles here. It refers to any angle that is between a right angle and a straight angle. Moreover, obtuse angles are also called as blunt angles.
What is the difference between acute and obtuse?
Acute angle has a smaller measure than right one, while obtuse angle has a bigger measure than right one.
Which shape has acute and obtuse angles?
Acute and obtuse triangles. An
What is less than a right angle?
Half a full rotation is called a straight angle, and a quarter of a full rotation is called a right angle. An angle less than a right angle is called an acute angle, an angle greater than a right angle (but less than a straight angle) is called an obtuse angle, and an angle greater than a straight angle (but less than a full angle | 677.169 | 1 |
Results don't match "similar triangles" definition
The definition for the "similar triangles" seems, wrong: the "reflection" assertion and following "scaling and reflection" have to be False.
Definition is:
1) ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
2) AB/XY = BC/YZ = AC/XZ
∠A, ∠B, ∠C are the angles of one triangle
∠X, ∠Y, ∠Z are the angles of another triangle
AB, BC, AC are the sides of one triangle
XY, YZ, XZ are the sides of another triangle | 677.169 | 1 |
5 Essential Aspects of Trigonometric Functions Mastery in Right Triangles
An Overview of Trigonometric Functions Mastery
Del
Understanding the Core Trigonometric Ratios
The essential trigonometric ratios—sine (sin), cosine (cos), and tangent (tan)—provide a powerful means to relate a right triangle's angles to the lengths of its sides. These basic yet profound connections are critical for progression into more complex scientific and engineering challenges.
The Sine Function: A Critical Ratio
Focusing on the sine function, we observe its purpose: to measure the ratio between the length of the side opposite an angle (θ) and the hypotenuse. This ratio is fundamental in various fields, including wave mechanics and circular motion.
Cosine: Partner to the Sine Function
Cosine is equally vital, associating an angle with the ratio of the adjacent side to the hypotenuse. Understanding cos(θ) is crucial for applications ranging from physics to astronomy, where the cosine function provides insight into planetary orbits and projectile dynamics.
Tangent: A Unique Proportion
Tangent stands out by directly contrasting the lengths of the opposite and adjacent sides. This unique ratio, tan(θ), is invaluable for calculating slopes and aiding in precise navigation.
The Pythagorean Identity's significance cannot be overstated as it seamlessly integrates the sine and cosine functions with the formula sin²(θ) + cos²(θ) = 1, showcasing their fundamental relationship.
Employing Trigonometric Functions has proven transformative in structural engineering, celestial navigation, and even sound engineering, where they assist in optimizing acoustics in designed spaces.
Calculating unknown measurements becomes straightforward with trigonometry. For example, the height of a mountain can be determined using the angle of elevation without physically ascending the peak.
Navigational Breakthroughs
Mariners and aviators rely heavily on Trigonometric Functions Mastery for accurate positioning and course plotting, revolutionizing how we traverse the world.
Moreover, the unit circle is a pivotal tool that links right-angle trigonometry to rotational movements, allowing for the interpretation of periodic phenomena across multiple disciplines.
Progressing to Complex Equations
Advancing further into the study of trigonometry, one encounters complex equations that intertwine various functions and angles. Tackling these advanced problems requires a deep understanding of trigonometric principles coupled with algebraic finesse.
Concluding Thoughts on Trigonometric Functions Mastery
Mastering trigonometric functions in right triangles transcends academic pursuits—it equips us with a versatile toolkit fostering innovation and problem-solving across numerous sectors. As we continue to delve deeper into this mathematical territory, the potential for its application appears infinite. | 677.169 | 1 |
Comparing flat Earth and spherical Earth from a geometric point of view
Go to page
Active Member
Here's the homemade test I suggested above. The idea is to observe the tip of the shadow of a stick etc. and mark its location every hour. The stick does not have to be straight or even completely vertical. The platform does not have to be completely horizontal either, but it must be flat for the follow-up test.
Picture 1 shows my version from a year ago. It had a miniature flagpole as a stick, the tip of which cast a shadow on the flat table. I marked the tip of the shadow with a marker on the table every hour.
Figure 2 shows the expected result of an experiment conducted at the vernal equinox. If you tighten the (red) threads from the tips of the shadows to the tip of the stick, you can see that the angles between the threads are all 15°. In addition to this, the tips of the shadow line up.
Especially the latter result is fatal for FE theory. The Sun rotating in a circle above the flat Earth cannot possibly produce such a straight line. The tip of the shadow should draw a circle.
(Admittedly, I did nothing but the latter test myself at that time, which succeeded as expected. I was too lazy to measure the angles. I hope you find an easy way to measure them.)
Figure 3 shows a similar test done on the summer solstice. In this case, the angles between the wires should be 13.76°. (You can probably be satisfied if you get a result of approx. 14°.) Now the tips of the shadow do not draw a straight line, but settle on an arc, which, strictly speaking, is a hyperbola.
Figure 4 shows an experiment done on the winter solstice. Even then, the angles between the threads are 13.76°. Even now, the tips of the shadow are placed on an arc, which is a hyperbola. This time it just opens in the opposite direction as in experiment 3. This is even more impossible in the FE model.
Active Member
You are right. I actually wanted to criticize a meme I found on the fb group "Official Flat Earth & Globe Discussion"
The meme was created by debunker Michel Toulouse (MT). He is a competent debunker, but this time his meme promises too much: "Anywhere on Earth, any season, all year long, any time of the day, a stick will show you the Sun appears to move 15° per hour."
However, the angle test shows that this only happens on the equinoxes. Well, blunders happen to debunkers too.
When I thought about it more, I began to have doubts. Now I think you can't measure the angles just from the pictures you get with the camera.
In the Toulouse meme, the angle between the tightened wires must be measured. In this case, the shape or position of the platform does not matter, nor does the shape or position of the stick (gnomon). This much is correct in the meme.
Let's consider situation A. In that case, the stick is straight and vertical. Suppose you have the camera high up exactly above the stick. The camera now sees shadows as straight lines, regardless of the shape of the platform. Let α be the angle between them in the photo. If you had installed the red wires, they would appear to be on top of the shadows, so in the camera image the angle between them is also α.
The angle α is the horizontal projection of the desired angle formed by the red wires (let it be β). This projection angle α is always bigger than the angle β. However, the angle β cannot be calculated if we do not know the height of the Sun. (E.g. in picture A, β = 15°. Let's assume that the height of the sun is 32°. I calculated that then α = 17.8°. I hope I calculated correctly.)
So I feel that it is not possible to determine the angle β from camera images alone. It has to be done by measuring the angle between the red wires manually (at least in the home test).
The question was surprisingly difficult for me and I am not at all sure that my explanation is correct. So if you do the test your own way, let us know how it went.
Senior Member.
Imagine being on a Pole. If you use a 90⁰ vertical gnomon here, on a flat level surface, you always get 15⁰ per hour for the shadows. This is because basically the sun stays where it is, maintaining its elevation, and the paper simply turns 15⁰ per hour as the Earth turns, for one 360⁰ circle per day.
An equatorial mount is motor-driven tripod for astronomical observations, where you set up the axis parallel to the Earth axis, and then set it to rotate 15.04⁰ per hour which makes it immobile with respect to the stars: basically the same as if you had set it up at the pole (until the ground gets in the way). Source: Note that the meme is still false.
Imagine summer solstice at the North Pole, piece of paper with a vertical gnomon, mark the shadows every hour, for 24×15⁰=360⁰ on the paper. However, the spatial angles are smaller (because the two sides enclosing the angle are longer) than the angles on the paper, so they are less than 15⁰. They only match on the equinox.
Active MemberTechnically the gnomon need not be perpendicular, but then it's not at the center of the circle. The shadow of the tip will move on a circle (as it is supposed to on FE if you don't tilt your paper), and measured from the center of that circle, you still get the 15⁰ per hour angle.
Using a gnomon aligned with the axis of the Earth ensures that it intersects the paper at that center.Active MemberThis is a horizontal sundial with the gnomon perpendicular to the hour disk. This sundial is dependent on your location. The hour disk is different at different latitudes. Only on the poles are the angles equal (15 degrees per hour). These hour discs work all year round, regardless of the time and direction of sunrise or sunset.
Let's apply my formula (1) to the angular displacement of the Sun in an hour. Then α = 15° (Earth's rotation per hour relative to the Sun). The angle β is the Sun's declination (the Sun's angular distance from the celestial equator).
The formula (1) gives the angle γ, which is the angular displacement of the Sun per hour as seen from Earth (the angle between the red wires). It varies depending on the season, so the Sun's declination β for that time must be known. On the equinoxes β = 0°, on the spring solstice β = 23.4° and on the winter solstice β = -23.4°.
The formula does not take the latitude into account. You get the same result no matter where on the globe you measure the angle between the red wires. Take the test tomorrow. I bet you get 15 degrees.
In this experiment we can forget how sundials work. The question is only about the size of the angle γ. My "artwork" below tries to show that the gnomon can be any shape (e.g. a crooked tree). The shape of the platform doesn't matter either (steep and uneven rock slope). The direction of the red arrows is the same regardless of these factors. And so is the angle γ.
I will try to do this string test tomorrow, if the weather in Helsinki permits. I'm sure I'll get a result of 15 degrees per hour. I also believe that you will get the same result no matter where you make the test.
Senior Member.Active Member
My formula calculates the angle γ for all days when we know the Sun's declination β on that day. In the table, the γ-angles are calculated for the period 20.3.2024 – 20.6.2024. Unfortunately, I don't know how to calculate the change in declination as a function of time (except on equinoxes and solstices). So I took the Sun's declinations directly from Stellarium. I used Helsinki time 05:14:00 as the spring equinox and continued every 24 hours from there.
I think my table is quite accurate for all places on earth. It can be pretty valid in other years as well, except maybe the last decimal.
The diagram shows the change in angle γ visually. It resembles a sine curve. At the moment the change is very slow, so you will probably get γ = 15° in the test for several more days. You can make the test at other times as well. For example, 25.5. according to the table, the result would be 14°.
"In the above formula, d is the number of days since January 1st (UTC 00:00:00). For example, On March 3rd (UTC 00:00:00), d = 31 + 28 + 2 = 61. On December 31st (UTC 00:00:00), d = 364." So March 20st is day number 78.
This formula does not take leap years into account. So this year March 20st is day number 79. When I took this into account and applied the formula, I got pretty much the same declinations as in my table (taken from Stellarium). The maximum difference was approx. 0.17°. The compatibility is quite good when we also remember that the equation assumes the earth orbits around the sun in a perfect circle.
My declination table above is probably more accurate than the values given by the formula because its values are "real time" given by Stellarium. So it probably gives quite accurate values also for the angle γ during this spring (for all places on earth). However, it cannot be applied to next year because the dates do not match. I probably also overestimated the accuracy of the table. There can be an error even in the first decimal place.
In any case, it is true that on equinoxes γ = 15° and on solstices γ = 13.76°. It can also be said that the change in angle γ roughly follows the shape of the graph below every year.
MemberMemberSenior Member. just like their challenges of finding water sticking to a sphere or gas pressure without a container and other examples that require a massive change of scale to demonstrate.
Furthermore, they don't seem to be bothered with effects that have no plausible mechanism, such as the Sun and Moon orbiting in circles above the flat Earth with absolutely no mechanism explaining why they do that. So, things like hurricanes spinning in certain directions could simply be a brute fact of the Earth, just like the orbiting Sun.
You cannot defeat Flat Earthers with reasoning. They do not appear to demonstrate scientific or mathematical reason beyond the simplest of platitudes, like their constant erroneous calls to "perspective" to explain away what obviously destroys their model.
Active MemberActive MemberYou are right. At the equinoxes, the sun rises exactly in the east and sets exactly in the west everywhere on Earth. Also, its altitude at noon is 90 minus the latitude of the location.
So, at the equator on the equinoxes, the sun rises exactly in east, is exactly at zenith at noon, and sets exactly in west 12 hours later. During all this time, the Sun's angular diameter is approx. 0.53 degrees.
Every flat-earther would have observed this a couple of days ago at the locality A (see picture below). But would he have believed his eyes? What should he have observed according to his own model?
According to the FE model, the sunrise in A means that the Sun suddenly appears as a small point (angular diameter approx. 0.19 degrees) in the north-eastern sky at an altitude of approx. 20 degrees. From there it continues its course until it is exactly at the zenith above point A at noon. In this case, its angular diameter is approx. 0.57 degrees. This is the same order of magnitude as in the ball model. This is the only moment that the observations and the FE model roughly match each other.
After that, the sun continues its journey along the equatorial circle at an altitude of 5000 km. The sunset as seen from A means that the Sun disappears from view in the northwest direction. When it disappears, the altitude is approx. 20 degrees and angular diameter approx. 0.19 degrees.
These sudden "disappearances" are explained in the FE theory, e.g. as follows: The sun is so far away that its light gets tired over a long distance. The Sun can only shine for a certain distance. Without such explanations, it is impossible to understand why the Sun is not visible at night. Even then, its altitude is about 14 degrees (seen from locality A).
The FE model and observations are in sharp contradiction. I haven't seen a single reasonable attempt to resolve this conflict – and I'm unlikely to see one.
I think the FE coffin is already firmly closed. The flat-earthers just don't believe it. Just as little would they believe their own eyes when looking at the sun's path at the equator.
Active MemberI made a "dynamic" model of the sun's paths on equinoxes and solstices for different latitudes. The model is easy to build. You basically just need a transparent hollow plastic ball, different colored markers and colored water.
Active Member
In the FE model, the Sun never really sets or rises, but always goes around a circular path above the flat Earth at an altitude of approx. 5000 km. Even at midnight the Sun's altitude is approx. 14.03° and angular diameter approx. 0.14°. Why don't we see it then though?
Active MemberIf GP || earth's axis, and GP _|_ PA and PB , then the plane PAB is the equatorial plane. Mendel's sketch therefore depicts an equatorial sundial, where the angle α is 15° per hour. The angle γ is also constant per hour, but generally γ < α. As Mendel said: "On the equinox, G=P and α=γ=15⁰."
Mendel's sketch can also be used to derive a general formula for the angle γ. In my sketch, the angle β is the Sun's declination (the Sun's angular distance from the celestial equator). This way I derived a new formula (2) for the angle γ. It's simpler than the formula (1) that I derived before, but it returns exactly the same values.
In a horizontal sundial, the hour disk is different for each latitude. However, this is not enough; the gnomon must be parallel to the earth's axis (as in the equatorial sundial). I didn't know this before. These customized sundials work all year round.
Here are the instructions for making a simple horizontal sundial: In the picture, the latitude is 20.46°. Would work e.g. in Tecolutla, Mexico or Vapi, India.
Active Member
Figure 1 shows the experiment I proposed above to determine the angle γ. In my test, the threads are attached slightly below the tip of the miniature flagpole. I marked the attachment point with a thin red plastic straw. This way, the "tip" of the shadow is a cross, which is more clearly readable (red dot in the picture).
I marked the "crosses" every hour with a marker on the surface of the table. Then I attached small nails at the marks and tightened the threads from them. The angle γ is now the angle between the wires.
According to the theory, on the equinox γ = 15°. However, I did the test only yesterday, 28.3. i.e. 8 days later. Then the angle should be (according to my calculation) slightly smaller, γ = 14.98°. It is worth repeating the test at the time of the summer solstice, when the angle should be clearly smaller, γ = 13.76°.
It was cloudy in Helsinki on the spring equinox and the cloudy weather just continued. Only yesterday 28.3. the sun was shining enough that I was able to make the test with the help of my friend Ilkka.
In the left picture, the "tips" of the shadows are already marked on the table. We got a clear mark at 10 a.m., 12 p.m. and 1 p.m. At 11 a.m. the sky was overcast, so that mark is missing. Fortunately, the marks 2, 3 and 4 were good enough for our measurements. In the pictures, Ilkka measures the angle between threads 2 and 3 with a protractor. In the picture on the right, you can see that the angle γ is approx. 15°. (Actually, it should be 14.98°, but of course our experiment is not capable of such accuracy.)
In addition, you can see from the picture on the left that the marks are no longer placed exactly on a straight line, as happens on equinoxes. After all, 8 days have already passed since the equinox.
Below is the measurement of the angle between threads 3 and 4. The result is still the same: the angle γ is approx. 15.
It is really worth repeating the experiment at the time of the summer solstice, when the angle γ is approx. 14° (more precisely γ = 13.76°). Even this modest test can clearly show the 1 degree difference. It also shows (at a qualitative level) that the "ball theory" works.
Senior MemberSenior Member.Surely ballistic drop is a far easier demonstration of the coriolis effect, not requiring a stochastic dynamic system? Just fire east and west near the equator - fire west and the target will rise and come towards you, fire east, and the target will drop and move away from you. Yes, unfortunately, those two effects are in opposite directions, but all you need to do is show that east and west behave differently (and differently from firing north or south - those two can also be used to show a sideways coriolis effect, but alas air movement is much more of a problem side to side than it is up and down, hence drop being the clearer indicator).
You'll probably need a decent fraction of a kilometer, or a large sample size - go on, you know you want to, in order to see the effect with statistical significance.
Note, this is all theoretical: all I know is the maths, not the hardware. It *should* be possible to demonstrate at the hunting-rifle scale, it doesn't just apply to big guns like the Paris GunSenior MemberThe UK is 0.04% of the planet, I'm not sure why you think it's important. I'm also curious which bit of the UK you think is at the equator.
And yes, snipers absolutely have to take the coriolis effect into account, at ranges much over a kilometer it makes a difference that can be measured in body widths. The planet is moving at a comparable speed to the muzzle velocity of a winchester 308 (about half) and the travel time is measured in seconds, it's significant.
Furthermore, they don't seem to be bothered with effects that have no plausible mechanism, such as the Sun and Moon orbiting in circles above the flat Earth with absolutely no mechanism explaining why they do that.
"Why" is a metaphysical question. Newton was eventually happy that the law applied - the attraction he could measure was that law enacting between the particles he was measuring.
And in Newton's 1713 General Scholium in the second edition of Principia: "I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses.... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies."[9]
Chasing the "why"s or the "causes" is a somewhat futile endeavour, as each time you unravel another layer you need to ask what causes *that*. If you begin that process, you have the choice of saying it just goes on for ever - which implies that there are more layers to the laws of physics than there are particles in a finite universe for those laws to act upon, or to accept that one of the layers is justified by "that's just how it is". The former seems paradoxical, and the latter is your ticket to just saying, as Newton did in 1713, that Newton's Law of Universal Gravitation "is enough". And to that extent, from his own perspective, it's fair to say that he did "know why" as much as anyone knowadays does. (This being a reference to @Mendel's "Isaac Newton didn't really know why, either." in #117 upthread.) But that's just my personal opinion on a metaphysical matter (what is "really" anyway?). Well-reasoned disagreement is fine on such things.
Senior Member.
This is why globes are useful for navigation while FE maps do not exist.
It's why we can use geometry to predict sunrise times and solar eclipses while FE can't.
And why there are always counterexamples to FE claims. | 677.169 | 1 |
Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2} - {a^2}{y^2} = 5$$, where a is the minimum distance between the curves y = ex and y = logex. Then $${a^2} + {1 \over {{b^2}}}$$ is equal to : | 677.169 | 1 |
ellipson
In the diagram, a circle of radius $OA=2$ rolls around the inside of a circle of radius $OB=4$ with the half radius $AP=1$ originally positioned so that $P$ is on $OB$. Find a parameterization for the path that $P$ traces out as the inner circle rolls (without slipping.) Start by choosing an appropriate parameter. Is that a familiar curve? | 677.169 | 1 |
Question
Since each corner of a
regular hexagon is a square
of a square drawn outside
the regular hexagon and.
connected by the vertices
of the adjacent squares,
let's figure out that
the resulting figure is a
regular polygon with 12
corners. | 677.169 | 1 |
Coplanar – Definition With Examples
Welcome to another exciting post from Brighterly, your trusted partner in learning mathematics in a fun, engaging, and understandable way. Today's topic is the concept of Coplanar, a fundamental aspect of geometry. This principle is essential to a wide range of studies, including engineering, computer graphics, and even architecture. Just as the world we live in is three-dimensional, understanding the properties of points and lines in planes is crucial to our understanding of space.
Throughout this article, we'll break down this seemingly complex concept into digestible parts. We'll start with the definition of coplanar and what it means in the world of geometry. From there, we'll explain the difference between collinear and coplanar, define coplanar and non-coplanar points, and explain coplanar and non-coplanar lines in geometry. Afterward, we'll guide you through how to determine whether four given points or two given lines are coplanar.
What Does Coplanar Mean in Geometry?
When you hear the term coplanar in geometry, it refers to points or lines that lie on the same geometric plane. These planes are invisible, flat surfaces that extend indefinitely in every direction. Even though we can't physically see these planes, they're a crucial part of understanding spatial relationships in mathematics. This concept is one of the key pillars of geometry, making it possible to visualize and solve complex geometric problems.
What Is the Difference Between Collinear and Coplanar?
Now, let's talk about collinearity and coplanarity. You might be wondering: are these just two fancy ways of saying the same thing? Not quite. Collinearity refers to the condition where three or more points lie on the same straight line. Coplanarity, on the other hand, allows for a broader range: points or lines can be in any arrangement as long as they're on the same plane.
What Are Coplanar and Non-coplanar Points?
When discussing coplanarity, we often encounter the terms coplanar and non-coplanar points. As we learned earlier, coplanar points lie on the same plane. For example, the four corners of a square are coplanar because they rest on the same flat surface. Conversely, non-coplanar points don't share a common plane. If you imagine a cube, the eight corners represent non-coplanar points because they occupy different planes.
What Are Coplanar and Non-coplanar Lines in Geometry?
Just like points, lines can be coplanar or non-coplanar too. Picture a notebook page filled with lines. All these lines exist on the same plane: the page surface. So, they are coplanar lines. Now, think about the lines that form the edges of a cube. They don't all lie on the same plane, which makes them non-coplanar lines.
Coplanar Points Definition in Geometry
Coplanar points in geometry are defined as three or more points that lie on the same plane. Remember, a plane is an infinitely large, flat surface. Therefore, no matter how far apart they are, if points share a plane, they're coplanar.
Non Coplanar Points Definition in Geometry
On the flip side, we have non-coplanar points. These are points that don't share a common plane. Imagine a pyramid. Each corner represents a point, and since these corners don't all lie on the same plane, they are non-coplanar points.
How to Determine Whether Given 4 Points are Coplanar?
It might sound complex, but determining if four points are coplanar isn't as tricky as it seems. You just need the coordinates of the points. We can determine coplanarity by checking the volume of the parallelepiped formed by the vectors of these points. If the volume equals zero, the points are coplanar.
Coplanar and Non Coplanar Lines
Again, we distinguish between coplanar and non-coplanar lines based on whether they lie on the same plane. The lines forming a square, for example, are coplanar. The edges of a cube, however, are non-coplanar lines because they exist on different planes.
How to Determine Whether Given 2 Lines are Coplanar?
To check if two lines are coplanar, you need their vector representations. If the determinant of their direction ratios is zero, the lines are coplanar. It might sound complex, but this concept becomes much easier with practice!
Condition For Coplanarity of Lines in Vector Form
The condition for coplanarity of lines in vector form is that the determinant of their direction ratios is zero. This is a key concept in vector algebra and helps solve many geometric problems.
Condition For Coplanarity of Lines in Cartesian Form
In Cartesian form, the condition for coplanarity is similar. If the determinant formed by the direction ratios of the lines equals zero, the lines are coplanar. This forms a basis for understanding many advanced geometry problems.
Important Notes on Coplanar
While we've covered a lot, here are some important notes on coplanar. Remember, any two points are always coplanar since they can exist on an infinite number of planes. Also, any three non-collinear points determine a unique plane, making them coplanar.
Solved Examples on Coplanar
Understanding the concept of coplanar is easier with practical examples. Let's look at some solved examples on coplanar to solidify your understanding.
Example 1: We have points A(1,2,3), B(4,5,6), C(7,8,9), and D(2,3,1). Are these points coplanar?
Solution: To find out if the points are coplanar, we calculate the volume of the parallelepiped formed by these points. We use the formula:
Conclusion
In this comprehensive guide from Brighterly, we have delved into the concept of coplanar, dissecting its definition, importance, and relevance in geometry, while also clarifying its difference from collinearity. We also provided some solved examples to better illustrate the concept and gave you some practice problems to test your understanding.
At Brighterly, we are committed to making learning mathematics as accessible, enjoyable, and interactive as possible. We hope this guide has helped you grasp the concept of coplanar and how it works in geometry. As with any other mathematical concept, the key to understanding coplanar is constant practice.
Remember, mathematics is not about memorizing formulas and methods; it's about understanding concepts and their applications. So, keep practicing, stay curious, and continue exploring the fascinating world of geometry with Brighterly!
Frequently Asked Questions on Coplanar
What are coplanar points?
Coplanar points refer to points that exist on the same geometric plane. In simpler terms, if you could connect the points with a flat sheet, without any point sticking out, they are considered coplanar.
How can I tell if points are coplanar?
To determine if points are coplanar, you can calculate the volume of the parallelepiped (a 3D shape) formed by vectors originating from these points. If the volume equals zero, then the points are coplanar.
Are any two lines always coplanar?
Not always. Two lines are coplanar if they exist on the same plane. This can occur if the lines are parallel, intersecting, or if they are the same line. However, if the lines are skew lines (non-parallel lines that don't intersect), they are not coplanar.
What's the difference between collinear and coplanar?
Collinear refers to points that lie on the same straight line, while coplanar refers to points that lie on the same geometric plane. Therefore, all collinear points are coplanar, but not all coplanar points are collinear Millions Thousands Hundreds Tens Ones 1 2 0 0 0 How to Write 1200000 in […]
12600 in Words
The number 12600 is written in words as "twelve thousand six hundred". It's six hundred more than twelve thousand. For instance, if you have twelve thousand six hundred stamps, you start with twelve thousand stamps and then add six hundred more. Thousands Hundreds Tens Ones 12 6 0 0 How to Write 12600 | 677.169 | 1 |
Similar presentations
Presentation on theme: "Kite , special member of Quadrilateral family"— Presentation transcript:
4 Hi, I am quadrilateralA quadrilateral is a four sided polygon. Therefore the total number of degrees of interior angles is 360°
5 I am TrapeziumA trapezoid is a quadrilateral with one pair of parallel lines.
6 I am parallelogramA parallelogram is quadrilateral comprised of two pairs of parallel lines.
7 A parallelogram is a quadrilateral whose opposite sides are parallel. Property: The opposite sides of a parallelogram are of equal length.
8 A parallelogram is a quadrilateral whose opposite sides are parallel. Property: The opposite angles of a parallelogram are of equal measure.
9 A parallelogram is a quadrilateral whose opposite sides are parallel. Property: The adjacent angles in a parallelogram are supplementary.
10 A parallelogram is a quadrilateral whose opposite sides are parallel. Property: The diagonals of a parallelogram bisect each other .
11 Hi, I am RectangleA rectangle is a parallelogram with four right angles. A rectangle is a parallelogram, its opposite sides must be congruent and it must satisfy all other properties of parallelograms .
12 I am RhombusA rhombus is a type of parallelogram, and what distinguishes its shape is that all four of its sides are congruent. There are several formulas for the rhombus that have
13 The diagonals of a rhombus are perpendicular bisectors of one another
15 A rhombus is a type of parallelogram, and what distinguishes its shape is that all four of its sides are congruent.Properties of the rhombus:all four sides are congruentDiagonals bisects each otherDiagonals are perpendicularA square is a rhombus and a rectangle. In other words, if each angle of a rhombus is 90°, then it's a square.
16 Hi, I am SquareA square has the properties of a rhombus and a rectangle. Its sides intersect at 90° and all four sides are congruent.
17 Kite Hi, I am Kite Who are you? Do you belong to family of Quadrilaterals?
18 Kite is a quadrilateral:. has two pairs of sides Kite is a quadrilateral: * has two pairs of sides. * each pair is adjacent sides (they meet) that are equal in length. Also, the angles are equal where the pairs meet. Diagonals meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other. | 677.169 | 1 |
87.
Óĺëßäá 16 ... xxvI . 1. ) DE = AB . PROP . XVII . 25. THEOREM . If the sides of any given equila- teral and equiangular figure of more than four sides , be produced so as to meet , the straight lines , joining their several intersections , shall | 677.169 | 1 |
HomeYear 1 NCETM ResourcesGeometry Recognise common 2D and 3D shapes presented in different orientations, and know that rectangles, triangles, cuboids and pyramids are not always similar to one another (3)
Recognise common 2D and 3D shapes presented in different orientations, and know that rectangles, triangles, cuboids and pyramids are not always similar to one another (3)
You can access our free resources by signing up to a free membership. Premium members can access our premium resources by signing up for a paid membership. You can sign up for a membership here.
Description
Pupils need a lot of experience in exploring and discussing common 2D and 3D shapes. In the process, they should learn to recognise and name, at a minimum:
rectangles (including squares), circles, and triangles
cuboids (including cubes), cylinders, spheres and pyramids.
Pupils need to be able to recognise common shapes when they are presented in a variety of orientations and sizes and relative proportions, including large shapes outside the classroom (such as a rectangle marked on the playground or a circle on a netball court). Pupils should be able to describe, using informal language (for example, "long and thin"), the differences between non-similar examples of the same shapes, and recognise that these are still examples of the given shape.
Pupils should practise distinguishing a given named shape type from plausible distractors. These activities should involve exploring shapes (for example, shapes cut from card) rather than only looking at pictures.
Additional information
Reviews
There are no reviews yet.
Be the first to review "Recognise common 2D and 3D shapes presented in different orientations, and know that rectangles, triangles, cuboids and pyramids are not always similar to one another (3EmailEmail | 677.169 | 1 |
The Elements of Euclid, Libros 1-6;Libro 11
Dentro del libro
Resultados 1-3 de 70 175 ... base HC be greater than the base CL , the triangle AHC is greater than the triangle ACL ; and if equal , equal ; and if less , less ; therefore as the base BC is to the base CD , so is the triangle ABC to the triangle ACD . [ V | 677.169 | 1 |
The content in this section builds on the content that was discussed in the previous section. Make sure that you have covered the content in the previous section before working through the content in this section (click here to be taken to the content in the previous section).
Bearing Main Rules From the previous section we learnt that there are 3 key rules that we need to remember with bearings. These rules are shown below:
They are measured from the North line
They are measured in a clockwise direction
They are measured in degrees and we always give them as 3 digits
Let's now have some more complex examples.
Example 1 The bearing of B from A is 135°. What is the bearing of A from B? The sketch below is not drawn to scale. Click here for a printable version.
We need to be really careful when we are answering questions like this and this is because the process to get to the answer is more complex than people expect.
We know that bearings are always measured clockwise from the north line. Therefore, the first step in answering this question is to draw a north line for point B.
The curved purple arrow on the above diagram is the bearing that we are looking for.
On the above diagram, we now have two north lines and these two lines will be parallel to one another. From the section on parallel lines, we know that the two angles inside parallel lines add up to 180°. This means that the angle that is 135° and the angle that is labelled x on the diagram below add up to 180°.
We are able to create the following equation from these two angles.
We want to find the values of x and we are able to do this by moving the 135 from the left side of the equation to the right. We move the 135 by doing the opposite; we take 135 from both sides of the equation.
Therefore, x is 45° and I have labelled this angle on the diagram below.
We are now in a position to find the size of the bearing. A full circle is 360°. This means that the bearing that we are looking for and the 45° angle will add up to 360°. We are able to create the following equation form this information; B in the equation below stands for the bearing that we are looking for.
We are able to find the value of B by moving the 45 from the left sides of the equation to the right. We move the 45 by doing the opposite; we take 45 from both sides of the equation.
Therefore, the bearing of A from B is 315°.
Example 2 There are 3 different points on the diagram below; point C, D and E.
Answer the following:
Find the bearing of C from D
Find the bearing of E from D
Part 1 The first part of this question is asking us to find the bearing of C from D. We answer this question in a very similar way to the previous question. The first step is to draw in a north line from D.
We always measure bearing clockwise from the north line. The bearing that we are looking for is shown on the diagram below (it is the green arrow).
We are able to find this bearing by finding the size of the angle that goes from the end of the bearing to the north line. I have labelled this angle y on the diagram below.
We can find the value of y by using the two parallel north lines. We know that angles inside parallel lines add up to 180°. We can create the following equation from this information.
We want to find the value of y, which we do by moving the 56 from the left to the right. We are able to do this by taking 56 from both sides of the equation.
y is 124°. I have added this angle to the diagram below.
We know that the bearing that we are looking for and the angle that we have just found will add up to 360°. Therefore, we can create the following equation (B in the equation below is the bearing that we are looking for):
We are able to find the value of B by taking 124 from both sides of the equation.
The bearing of C from D is 236°.
Part 2 The second part asks us to find the bearing of E from D. The bearing that we are looking for is shown on the diagram below.
On the above diagram we are given a south line. The south line will have a bearing of 180°. There are two angle that make up the south line bearing; these angles are the bearing that we are looking for in this question (the bearing of E from D) and the angle that we are given in the diagram (the angle that is 48°). Therefore, we can create the following equation from this information (I am going to let the bearing that we are looking for equal B):
We want to find the value of B, and we do this by taking 48 from both sides of the equation. | 677.169 | 1 |
How to Find the Exact Value of a Trig Function – Quick and Easy Guide
To find the exact value of a trigonometric function, I first consider the specific angle in question. Some angles, like $30^\circ$, $45^\circ$, and $60^\circ$, have well-known exact values for sine, cosine, and tangent functions, which are derived from specific right triangles.
I refer to the unit circle, where the circumference represents angles in radians, and position coordinates correspond to the values of the trigonometric functions at that angle.
Understanding the symmetries and periodic properties of the trigonometric functions allows me to determine exact values for a wider range of angles.
By applying certain identities, such as the Pythagorean identity or angle sum and difference formulas, I can find the trigonometric values for other angles not immediately found on the unit circle.
Steps Involved in Finding the Exact Values for Trigonometric Functions
To find the exact values for trigonometric functions such as sin, cos, and tan, I follow a structured approach. Here's how I do it:
Identify the Angle: First, I determine the angle in degrees or radians. If necessary, I convert between degrees and radians using the formula: $$ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} $$
Reference Angles: I find the reference angle, which is the acute angle that the terminal side of my given angle makes with the x-axis. This simplifies calculations, especially for angles greater than $90^\circ$ (or $\frac{\pi}{2}$ radians).
Quadrants and Signs: The sign of a trigonometric function depends on which quadrant the angle lies in:
Quadrant
sin
cos
tan
First
+
+
+
Second
+
–
–
Third
–
–
+
Fourth
–
+
–
Special Angles and Unit Circle: I consider special angles that have known exact values on the unit circle, such as $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and v90^\circ$; or in radians: (0), $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$. The coordinates of these angles on the unit circle correspond to the values of cos and sin, respectively.
By using a combination of these approaches, I effectively find exact trigonometric values for angles. Remember, practice helps to reinforce these concepts and improve the speed and accuracy of calculations.
Practical Tips and Tools
Finding the exact values of trigonometric functions can be simplified by using a combination of practical tools and memorization techniques. Here's how I approach it:
Calculators: A scientific calculator is an indispensable tool for checking work. Make sure it's set to the correct mode (degrees or radians) to match the problem you are working on.
Memorization: I find it useful to commit to memory the values of sine, cosine, and tangent for common angles such as $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$, and $\frac{\pi}{2}$. This is a small list that can significantly speed up calculations.
Angle (Radians)
$\sin$
$\cos$
$\tan$
( 0 )
( 0 )
( 1 )
( 0 )
$\frac{\pi}{6} $
$ \frac{1}{2} v
$\frac{\sqrt{3}}{2}$
$ \frac{1}{\sqrt{3}} $
$\frac{\pi}{4}$
$\frac{\sqrt{2}}{2}$
$\frac{\sqrt{2}}{2}$
( 1 )
$\frac{\pi}{3} $
$ \frac{\sqrt{3}}{2}$
$\frac{1}{2}$
$\sqrt{3}$
Articles and Lists: I often refer to articles and lists online that provide the trigonometric equations and properties. This is useful when dealing with more complex problems in algebra and calculus, especially when exploring the domains of these functions.
Algebraic Techniques: Familiarity with algebraic manipulations can also help solve for trigonometric values. I work through algebraic expressions using identities like the Pythagorean identity $ \sin^2(x) + \cos^2(x) = 1 $ or angle sum and difference formulas.
Calculus Applications: For those exploring calculus, understanding the integrals and derivatives of trigonometric functions is crucial. This knowledge often provides insights into the behavior of these functions across different domains.
Conclusion
In this guide, I walked through the process needed to find the exact values of trigonometric functions. To recap, let's consider the essential steps for achieving such values:
As you become more familiar with these strategies, your ability to find the exact trigonometric values will improve. Be sure to practice with a variety of angles and problems to build your confidence and proficiency.
Remember, understanding the geometrical interpretations of these functions can also immensely clarify their exact values.
By consistently using these methods, you'll be able to swiftly and accurately determine the trigonometric values you need, whether for academic purposes, problem-solving, or real-world applications. | 677.169 | 1 |
Question 3.
If E, F, G and H are respectively the midpoints of the sides AB, BC, CD and AD of a parallelogram ABCD, show that ar (EFGH) =1/2 ar (ABCD).
Solution:
Given that □ABCD is a parallelogram.
E, F, G and H are the midpoints of the sides.
Join E, G.
Now
ΔEFG and □EBCG he on the same base EG and between the same parallels
EG // BC.
∴ ΔEFG =1/2□EBCG ……………(1)
Similarly,
ΔEHG =1/2□EGDA …………….(2)
Adding (1) and (2);
ΔEFG + ΔEHG =1/2□EBCG +12□EGDA
□EFGH =1/2[□EBCG +□ EGDA]
□EFGH =1/2[□ABCD]
Hence proved.
Question 4.
What figure do you get, if you join ΔAPM, ΔDPO, ΔOCN and ΔMNB in the example 3 ?
Solution:
□ABCD is a rhombus.
M, N, O and P are the midpoints of its sides. By joining ΔAPM, ΔDPO, ΔOCN and ΔMNB we get the figure shown by shaded region.
Question 5.
P and Q are any two points lying on the sides DC and AD .respectively of a parallelogram ABCD. Show that ar (ΔAPB) = ar (ΔBQC).
Solution:
ΔAPB and □ABCD are on the same base
AB and between the same parallel lines
AB//CD.
∴ ΔAPB =1/2□ABCD …………… (1)
Also ΔBCQ and □BCDA are on the same base BC and between the same paral¬lel lines BC//AD.
∴ ΔBCQ =1/2□BCDA …………….. (2)
But □ABCD and □BCDA represent same parallelogram.
∴ΔAPB = ΔBCQ [from (1) & (2)]
Question 6.
P is a point in the interior of a parallelogram ABCD. Show that
i) ar (ΔAPB) + ar (ΔPCD) = 1/2 ar(ABCD)
(Hint : Through P, draw a line paral¬lel to AB)
Question 7.
Prove that the area of a trapezium is half the sum of the parallel sides mul¬tiplied by the distance between them.
Question 8.
PQRS and ABRS are parallelograms and X is any point on the side BR.
Show that
i) ar (PQRS) = ar (ABRS)
Solution:
□PQRS and □ABRS are on the same base SR and between the same parallels SR//PB.
∴ □PQRS = □ABRS
ii) ar (ΔAXS) =1/2 ar (PQRS)
Solution:
From (1) □PQRS = □ABRS
And □ABRS and ΔAXS are on the same base AS and between the same paral¬lels AS//BR.
∴ ΔAXS =1/2□ABRS
=1/2□PQRS from (1)
Hence proved.
Question 9.
A farmer has a held in the form of a parallelogram PQRS as shown in the figure. He took the midpoint A on RS and joined it to points P and Q. In how many parts the field is divided ? What are the shapes of these parts ? The farmer wants to sow groundnuts which are equal to the sum of pulses and paddy. How should he sow ? State reasons.
Solution:
From the figure ΔAPQ, □PQRS are on the same base PQ and between the same parallels PQ//SR.
∴ ΔAPQ =12□PQRS
⇒ □PQRS – AAPQ =12□PQRS
∴12□PQRS = ΔASP + ΔARQ
∴ The farmer may sow groundnuts on ΔAPQ region.
The farmer may sow pulses on ΔASP region.
The farmer may sow paddy on ΔARQ region.
Question 10.
Prove that the area of a rhombus is equal to half of the product of the diagonals.
Question 1.
In a triangle ABC, E is the midpoint of median AD. Show that
Question 4.
In the figure ΔABC; D, E and F are the midpoints of sides BC, CA and AB respectively. Show that
i)BDEF is a parallelogram
ii) ar (ΔDEF) =1/4 ar(ΔABC)
iii) ar (BDEF) =1/2 ar(ΔABC)
Solution:
i) In ΔABC; D, E and F are the mid¬points of the sides.
∴ EF//BC FD//AC ED//AB
EF =1/2 BC FD =1/2 AC ED =1/2 AB
[ ∵ line joining the mid points of any two sides of a triangle is parallel to third side and equal to half of it]
∴ In □BDEF
BD = EF [ ∵ D is mid point of BC and12BC = EF]
DE = BF
∴ □BDEF is a parallelogram.
Question 9.
In the figure, if ar (ΔRAS) = ar (ΔRBS) and ar (ΔQRB) = ar (ΔPAS) then show that both the quadrilaterals PQSR and RSBA are trapeziums.
ΔRAS = ΔRBS ………….. (1)
Both the triangles are on the same base RS and between the same pair of lines RS and AB.
As their areas are equal RS must be parallel to AB.
⇒ RS//AB
∴ □ABRS is a quadrilateral in which AB//RS.
∴ □ABRS (or) □RSBA is a trapezium.
Now AQRB = APAS (given)
⇒ ΔQRB – ΔRBS = ΔPAS - ΔRAS
[from (1) ΔRBS = ΔRAS]
⇒ ΔQRS = ΔPRS
These two triangles are on the same base RS and between the same pair of lines RS and PQ.
As these two triangles have same area RS must be parallel to PQ.
⇒ RS // PQ
Now in quad PQRS; PQ//RS.
Hence □PQRS is a trapezium.
Question 10.
A villager Ramayya has a plot of land in the shape of a quadrilateral. The grampanchayat of the village decided to take over some portion of his plot from one of the comers to construct a school. Ramayya agrees to the above proposal with the condition that he should be given equal amount of land in exchange of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will implemented. (Draw a rough sketch of the plot.)
Solution:
Let □ABCD is the plot of Ramayya.
School be constructed in the region ΔMCD where M is a point on BC such that □ABCD ≅ ΔADE
Draw the diagonal BD.
Draw a line parallel to BD through C which meets AB produced at E.
Join D, E
ΔADE is the required triangle. | 677.169 | 1 |
a can range from 45° to 60°, which of the following are possible va
[#permalink]
10 Dec 2022, 12:27
Since angle a has a minimum of 45 degrees and a maximum of 60 degrees, then the y coordinate will a range of values between the minimum and maximum angles.
Let's fine the minimum first:
Attachment:
45 degrees.jpg [ 17.88 KiB | Viewed 1042 times ]
If angle a = 45 degrees, then we have special 45-45-90 isosceles triangle with sides 4-4-4 sq root 2. That means in this case y = 4
Now for the maximum:
Attachment:
60 degrees.jpg [ 18.73 KiB | Viewed 1017 times ]
If angle a = 60 degrees, then we have another special triangle, a 30-60-90 triangle. This has the unique properties of having a side length ratio of (small leg) : (larger leg) : (hypotenuse) = x : x square root 3: 2x.
That means y = 4 square root 3 which is approximately equal to 6.9
So now that we have the minimum and maximum values for y, we can set up an inequality:
Re: If a can range from 45° to 60°, which of the following are possible va
[#permalink]
11 Dec 2022, 06:00
Expert Reply
OE
Given that the angle ranges from 45° to 60°, you need to plug in values for angle a and find a special triangle to solve for y. If a is 45°, the triangle's sides are \(x, x, x \sqrt{2}\). It doesn't matter what the hypotenuse is; x = 4, which means y also is 4. If a is 60°, the triangle's sides are \(x, x \sqrt{3}, 2x\). The shortest side of the triangle would be the one on the x-axis. Since x = 4, then \(y = 4 \sqrt{3}\) or approximately 6.93. So the correct answers range from 4 to 6.93. Choices (C), (D), and (E) are all correct.
_________________ | 677.169 | 1 |
which constructs the hyperbolic bisector of two points p and q lying in the Poincaré disk. The endpoints of the resulting hyperbolic segment lie on the circle at infinity. It must also provide the function operator
where the points p, q, and r lie in the Poincaré disk. This overloaded version constructs the hyperbolic bisector of the segment [p,q] limited by the hyperbolic circumcenter of p, q, r on one side and the circle at infinity on the other. Moreover, it must provide the function operator
where the points p, q, r, and s lie in the Poincaré disk. This overloaded version constructs the hyperbolic bisector of the segment [p,q] limited by the hyperbolic circumcenter of p, q, r on one side, and the hyperbolic circumcenter of p, s, q on the other side.
In the Poincaré disk model, a hyperbolic segment is supported either by the Euclidean circle that passes through the two points and is perpendicular to the circle at infinity, or by the Euclidean line that passes through the two points and the origin. Abusively, we allow one or both endpoints of the segment to lie on the circle at infinity, so a hyperbolic segment can actually represent a hyperbolic ray or a hyperbolic line.
which returns a boolean indicating whether the triangle defined by the points p0, p1, and p2 is hyperbolic (i.e., if its circumscribing disk is contained in the unit disk). It must also provide the function operator
which returns whether the triangle is hyperbolic, and if not stores in ind the index of the non-hyperbolic edge of the triangle, as defined in [1]. The edge of the triangle opposite to pj for j = 0,1,2 is considered to have index j.
Oriented_side operator()(Hyperbolic_point_2 p, Hyperbolic_point_2 q, Hyperbolic_point_2 r, Hyperbolic_point_2 t), which returns the position of the point t relative to the oriented circle defined by the points p, q, and r.
Oriented_side operator()(Hyperbolic_point_2 p, Hyperbolic_point_2 q, Hyperbolic_point_2 query), which returns the position of the point query relative to the oriented hyperbolic segment with vertices p and q. | 677.169 | 1 |
If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. 730 mzacb=mab an inscribed angle is an angle that has its vertex on the edge of the circle extending inward to the opposite edge of the circle.
Source: mgeo.weebly.com
Problems involve central angles and inscribed angles. This worksheet will help you explore the relationship between inscribed angles and their corresponding central angles.
Source: getunnelt.blogspot.com
If angle acb is 47 degrees, what is the measure. Central and inscribed angles a central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle.
Source: hualahdalah.blogspot.com
Math10 tg u2 from central angles and inscribed angles worksheet answer key source. 5 derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality;
Source: gianueamin10.blogspot.com
Inscribed angles date period state if each angle is an inscribed angle. The arc formed by the intersection of the two sides of the angle and the circle is called an
1) ∠Fqe F E D Q 2) ∠1 H I J 1 Name The Central Angle Of The Given Arc.
For each figure, determine the indicated measures. Its measurement of a diameter, and area and sell original idea from. What arc is intercepted by ∠ ?
Inscribed Angle Abc Is Also Described.
The Arc Formed By The Intersection Of The Two Sides Of The Angle And The Circle Is Called An
Central and inscribed angles a central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle. 730 mzacb=mab an inscribed angle is an angle that has its vertex on the edge of the circle extending inward to the opposite edge of the circle. For the circle at right with center c, ∠acb is a central angle.
What Is A Shape Is An Answer Key Idea Of That Intercept The Outside Of The Distance Learning Moderate Can Do A Copy Of Answers And Arcs Central Angles Inscribed Worksheet Worksheets Includes.
5 derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; Math10 tg u2 from central angles and inscribed angles worksheet answer key source. Angles outside circie 5 pack my father retired years ago for a natural gas | 677.169 | 1 |
How Do You Tell If It Is An Ellipse Or Hyperbola?
If the squared x term and the squared y term are opposite signs (one is positive and one is negative), then you have a hyperbola. If the squared x term and the squared y term have the same constant multiplier (for example, 3x2 + 3y2), then you have a circle. The only other choice is an ellipse.
how do you tell the difference between a circle and an ellipse equation?
The only difference between the circle and the ellipse is that in an ellipse, there are two radius measures, one horizontally along the x-axis, the other vertically along the y-axis. Clearly, for a circle both these have the same value. By convention, the y radius is usually called b and the x radius is called a.
If the intersection point is double, the line is a tangent line. Intersecting with the line at infinity, each conic section has two points at infinity. If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola.
For ellipses, a≥b (when a=b , we have a circle) a represents half the length of the major axis while b represents half the length of the minor axis.
What does an ellipsis look like Some writers and editors feel that no spaces are necessary. You may also read, How do you tell if it is exothermic or endothermic?
What is ellipse and its properties?
An ellipse is formed by a plane intersecting a cone at an angle to its base. All ellipses have two focal points, or foci. The sum of the distances from every point on the ellipse to the two foci is a constant. All ellipses have a center and a major and minor axis. Check the answer of How do you tell if Michael Kors jacket is authentic?
How is a hyperbola formed?
A hyperbola is formed by the intersection of a plane perpendicular to the bases of a double cone. All hyperbolas have an eccentricity value greater than 1 . All hyperbolas have two branches, each with a vertex and a focal point.
What is standard form of an ellipse?
The equation of an ellipse in standard form. The center is (h,k) and the larger of a and b is the major radius and the smaller is the minor radius. follows: (x−h)2a2+(y−k)2b2=1. The vertices are (h±a,k) and (h,k±b) and the orientation depends on a and b. Read: How do you tell if my wheel bearing or ball joint is bad?
What does a hyperbola look like?
A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.
Is a circle an ellipse?
In fact a Circle is an Ellipse, where both foci are at the same point (the center). In other words, a circle is a "special case" of an ellipse. Ellipses Rule!
Does an ellipse have a radius?
The radius is the line from the center of an object to its perimeter. An ellipse, which is like a circle that has been elongated in one direction, has two radii: a longer one, the semimajor axis, and a shorter one, the semiminor axis.
Is a parabola a hyperbola?
Parabola vs Hyperbola. When a set of points in a plane are equidistant from a given directrix or a straight line and from the focus then it is called a parabola. When the difference of distances between a set of points present in a plane to two fixed points is a positive constant, it is called a hyperbola.
What are hyperbola asymptotes?
Asymptotes are imaginary lines that a function will get very close to, but never touch. The asymptotes of a hyperbola are two imaginary lines that the hyperbola is bound by. It can never touch the asymptotes, thought it will get very close, just like the definition of asymptotes states.
What is the use of hyperbola?
When two stones are thrown in a pool of water, the concentric circles of ripples intersect in hyperbolas. This property of the hyperbola is used in radar tracking stations: an object is located by sending out sound waves from two point sources: the concentric circles of these sound waves intersect in hyperbolas. | 677.169 | 1 |
Series of superimposed regular polygons
In summary, the conversation discusses the formation of a sequence of concentric regular polygons with equal area and maximal symmetry, starting with an equilateral triangle and approaching a circle. The formula for the radius of each n-gon is given, and the formula for the fraction of the area not occupied by any successive polygons is derived using trigonometry. The conversation also explores the effect of rotation and the possibility of deriving a unique series or fundamental constant from the problem.
Feb 25, 2007
#1
Loren Booda
3,125
4
Superimpose concentric regular polygons of equal area with maximal symmetry, starting with the equilateral triangle and sequentually approaching the circumference of a circle. What series can you derive for the fraction of the area not occupied by any successive polygons?
Okay:
Now, clearly we can form a radius sequence for each n-gon, where the radius for each n-gon [itex]R_{n}[/itex] is given by the formula:
[tex]R_{n}=\sqrt{\frac{2A}{n\sin(\frac{2\pi}{n})}}[/tex]
This value is probably needed to solve your problem in some manner.
Assuming you mean a sequence of polygon inscribed in a circle of radius R, each n-gon can be interpreted as n isosceles triangle with congruent sides of length R and angle between them of [itex]2\pi/n[/itex] which can then be divided into two right angles with angle [itex]\pi/n[/itex]. The base of each such triangle is [itex]2R sin(\pi/n)[/itex] and the height is [itex]R cos(\pi/n)[/itex] so the area of each triangle is [itex]R^2 sin(\pi/n) cos(\pi/n)[/itex] and the area of the entire n-gon is [itex]nR^2 sin(\pi/n) cos(\pi/n)[/itex].
Since you are asking about the area inside the circle NOT in the polygon, that would be [itex]\pi R^2- nR^2 sin(\pi/n) cos(\pi/n)[/itex] and the fraction of the area of the circle not occupied by the n-gon would be
[tex]\frac{\pi- n sin(\pi/n) cos(\pi/n)}{\pi}= 1-\frac{n}{\pi} sin(\pi/n)cos(\pi/n)[tex].
It's easy to see that the last term of that goes to 1 in the limit and the "fraction of the area of the circlee not occupied by the n-gon", of course, goes to 0.
I'm not sure what you mean by "fraction of the area not occupied by successive polygons".Feb 25, 2007
#5
Loren Booda
3,125
4
Sorry, by "a circle" I meant that a sequence of regular polygons of equal area and n sides, as n approaches infinity, approaches a circle of equal area.
arildno's formula, in an infinite series, might be used to determine my sequence - the fractional areas of regular polygons that are not included within successively sided, concentric, and bilaterally symmetric regular polygons of equal area.
I believe his second post captures the gist of what I am proposing.
HallsofIvy, would you repost your last formula?
Last edited: Feb 25, 2007
Feb 25, 2007
#6
ramsey2879
841
3
arildno said:If a coaxial N+1 sided polygon is rotated with respect to the previous N sided regular polygon of the same area, would the amount of the N sided polygon that is not covered be changed?
It was supposed to be
[tex]\frac{\pi- n sin(\pi/n) cos(\pi/n)}{\pi}= 1-\frac{n}{\pi} sin(\pi/n)cos(\pi/n)[/tex].
Feb 26, 2007
#11
ramsey2879
841
3
Loren Booda said:
I believe rotation would affect the areas in question - that's why I asked for maximal symmetry - e. g., all polygons each resting on a side.
You also said concentric polygons, so I take your posts to specify that each polygon has the same center axis and that a perpendicular bisector of the bottom side the n+1 polygon coincides with a perpendicular bisector of the bottom side of the previous n sided polygon, though the distance to the bottom side is shorter with each subsequent polygon.
Still I have difficulty comming up with a plot of the respective polygons in polar coordinates. Is it sufficient for your purpose to just add up the area of the N-sided polygons that lie outside the radius of a circle of the same area?
Last edited: Feb 27, 2007
Feb 26, 2007
#12
Loren Booda
3,125
4
I was trying to discern whether a unique series or fundamental constant could be derived from the problem at hand. It seems now that it is merely an exercize in excruciating geometry.
Some derivation of HallsofIvy's formula would probably do the trick, though. Thanks all for your patience.
What is a series of superimposed regular polygons?
A series of superimposed regular polygons is a set of polygons that are stacked on top of each other in a way that each subsequent polygon shares a common side with the previous one. These polygons are all regular, meaning that all their sides and angles are equal.
What is the purpose of creating a series of superimposed regular polygons?
The purpose of creating a series of superimposed regular polygons is to demonstrate the concept of tessellation, which is the process of filling a plane with repeating shapes without any gaps or overlaps. This concept is important in mathematics, art, and science.
What are some real-life examples of series of superimposed regular polygons?
Some real-life examples of series of superimposed regular polygons include tiled floors and walls, honeycomb structures in beehives, and geometric patterns in Islamic art and architecture.
What is the relationship between the number of sides in each polygon and the overall shape of the series?
The relationship between the number of sides in each polygon and the overall shape of the series is that as the number of sides increases, the series will appear more circular and smooth. This is because the polygons are able to fit together more closely without any gaps or overlaps.
How is the concept of series of superimposed regular polygons related to other mathematical concepts?
The concept of series of superimposed regular polygons is closely related to other mathematical concepts such as symmetry, geometry, and patterns. It also has applications in fields such as architecture, engineering, and computer graphics. | 677.169 | 1 |
How to Prove Triangles Similar – Quick Guide
In geometry, triangles are fascinating shapes that come with a set of intriguing properties. One such property is similarity, which allows us to compare triangles based on their shapes, regardless of their sizes. Proving triangles similarly is a fundamental skill in geometry that has practical applications in various fields, including engineering, architecture, and even art.
What Does Similarity Mean in Geometry?
Similarity is like a secret code that tells us when two figures are essentially the same shape, even if they're not the same size. It's like looking at two photos of the same object, one zoomed in and one zoomed out. They're the same thing, just at different scales.
When we talk about similar triangles, we're saying that they're like geometric twins. Their corresponding angles match up perfectly, so they have the same "face," so to speak. But it's not just about the angles. The sides of these triangles are in proportion, which means if you were to measure them, you'd find that the lengths are scaled versions of each other. It's like one triangle is a mini-me or a giant version of the other.
Image: collegedunia.com
Imagine you have two triangles, and you can magically stretch or shrink one without changing its shape. If you can make it match the other triangle exactly, then they're similar. This concept of similarity is super useful because it lets us compare shapes, solve problems, and understand the world around us in a more mathematical way. So, next time you see two shapes that look alike but are different sizes, remember, they might just be similar!
What Are the Triangle Similarity Criteria?
Proving that triangles are similar is like solving a puzzle. We use specific criteria, or "clues," to figure out if two triangles are basically the same shape. These criteria are all about how the angles and sides of the triangles relate to each other. Let's dive into the three main criteria for triangle similarity:
Angle-Angle (AA) Similarity: This criterion is all about the angles. If two angles in one triangle are exactly the same as two angles in another triangle, then the triangles are considered similar. It's like if two people have the same eye color and smile – you can tell they're related! In geometry, if two triangles share two congruent angles, the third angle must be congruent too, since the angles in a triangle always add up to 180 degrees. So, by knowing just two angles are the same, we can say the whole triangle is similar.
Side-Side-Side (SSS) Similarity: This one is about the sides. If the lengths of the corresponding sides of two triangles are in the same proportion, then the triangles are similar. For example, if one triangle has sides that are twice as long as the sides of another triangle, they're similar. It's like comparing two photographs of the same object taken from different distances – they show the same thing, just at different scales.
Side-Angle-Side (SAS) Similarity: This criterion combines sides and angles. If two sides of one triangle are in the same proportion as two sides of another triangle, and the angles between those sides are equal, then the triangles are similar. It's like saying if two people have the same height-to-weight ratio and the same nose shape, they have a similar appearance.
Image: mathmonks.co
Three Accepted Methods for Proving Triangles Similar
Based on these three criteria for triangles similarity, there are therefore 3 methods to prove such similarity.
AA Similarity
Show that two angles of one triangle are congruent to two angles of another triangle. Since the sum of the angles in a triangle is always 180 degrees, if two angles are congruent, the third angle must also be congruent, proving the triangles are similar.
We have ∠A = ∠D and ∠B = ∠E. Since the third angles ∠C and ∠F are also equal, the triangles are similar by AA similarity (AAA is a special case of AA).
SSS Similarity
Demonstrate that the ratios of the corresponding sides of two triangles are equal. This can be done by dividing the lengths of corresponding sides and showing that the ratios are the same for all three pairs of sides.
Prove that two sides of one triangle are proportional to two sides of another triangle and that the angles between these sides are congruent. This method combines the properties of proportional sides and congruent angles to establish similarity.
Problem: Prove that triangles MNO and PQR are similar given that MN/PQ = 3/4, NO/QR = 3/4, and ∠N = ∠Q.
Solution:
We are given that the ratios of two pairs of corresponding sides are equal: MN/PQ = NO/QR = 3/4.
We are also given that the included angles are congruent: ∠N = ∠Q.
Therefore, triangles MNO and PQR are similar by SAS similarity.
Finding Missing Side Lengths in Similar Triangles
Once you've established that two triangles are similar, you unlock a powerful tool for finding missing side lengths: the concept of proportional sides. Similar triangles maintain the same shape, which means their corresponding sides are proportional. This proportionality is governed by a scale factor, which we'll call k.
Let's say you have two similar triangles, Triangle 1 and Triangle 2. The scale factor k describes how much larger or smaller Triangle 2 is compared to Triangle 1. Mathematically, this relationship is expressed as:
or, equivalently,
This equation tells us that if you multiply the length of a side in Triangle 1 by �k, you'll get the length of the corresponding side in Triangle 2, and vice versa. Now, imagine you know the lengths of some sides but there's one side length missing. You can set up a proportion using the known side lengths and the scale factor to find the unknown length. Here's how you do it:
Set up the proportion: Write an equation that relates the known side lengths and the unknown length using the concept of proportional sides.
Cross-multiply: Multiply the terms diagonally across the equal sign. This will give you an equation where the unknown length is multiplied by a known number.
Solve for the unknown: Use algebraic manipulation to isolate the unknown length on one side of the equation. This might involve dividing both sides of the equation by a number or subtracting a number from both sides.
By following these steps and using the principle of proportional sides, you can confidently find missing side lengths in similar triangles, unlocking a world of geometric possibilities!
Conclusion
Proving triangles similar is a crucial aspect of geometry that allows us to understand the relationships between different shapes. By mastering the triangle similarity criteria and learning how to apply them, you can unlock a world of geometric possibilities. Whether you're solving complex problems or simply exploring the beauty of shapes, the ability to prove triangles similar is an invaluable tool in your mathematical toolkit.
FAQ
What are the criteria for proving triangles similar?
To prove that triangles are similar, we use specific criteria that compare the angles and sides of the triangles. The three main criteria are:
If two angles in one triangle are equal to two angles in another triangle, then the triangles are similar. This is because if two angles are the same, the third angle must also be the same, making the triangles have identical shapesCan triangles be similar but not congruent?
Yes, triangles can be similar but not congruent. Similar triangles have the same shape but may be different in size, while congruent triangles are identical in both shape and size. For example, two triangles with angles of 30°, 60°, and 90° are similar because they have the same shape, but if one has sides of 3, 4, and 5 units and the other has sides of 6, 8, and 10 units, they are not congruent because their sizes are different.
How do I find missing side lengths in similar triangles?
To find missing side lengths in similar triangles, you use the fact that corresponding sides are proportional. Write a ratio that compares the lengths of the known sides in one triangle to the lengths of the corresponding sides in the other triangle. Make sure your proportion includes the missing side length. For example, if you know sides a and b in one triangle and side c in the other triangle, but side d is missing, your proportion might look like a/b = c/d. Use cross-multiplication to solve the proportion for the missing side length. This involves multiplying the numerator of one ratio by the denominator of the other ratio and setting the two products equal to each other. Once you've found the missing length, double-check your work to make sure it makes sense in the context of the problem | 677.169 | 1 |
...the first and second terms AABD, we have AABC : ADBE=AB'BC : DB-BE, (Alg. 116) Cor. Parallelograms which have one angle of the one equal to one angle of the other, are to one another in the ratio which is compounded of the ratios of the sides about the equal angles....triangles ABC, DEF are equiangular : wherefore, if the sides, &c. PROP. VI. THEOR. — If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals; the remaining angles are equal, each to each, viz....
...equiangular to the triangle DEF. Wherefore, if the sides, &c. QED PROPOSITION VI. THEOB. — If two triangles have one angle of the one equal to one angle of the oiher, and the sides about the equal angles proportionals, the triangles shall be equiangular, and...
...equiangular, and shall have those angles equal about which the sides are proportionals. Let ABC, DEF be two triangles, which have one angle of the one equal to one angle of the other, viz. the angle BAC to the angle EDF, and the sides about two other angles ABC, DEF, proportionals,...
...as K to L ; and therefore ex oequali, BD will be to CG, as I to L (by Prop. 33, B. 5). COR. 2. — Triangles which have one angle of the one equal to one angle of the other, are to each other in a ratio compounded of the ratios of the sides about the equal angles. COR. 3....
...angle of the other, have their sides about the equal angles reciprocally proportional : arallelograms which have one angle of the one equal to one angle of And par the other, and their sides about the equal angles reciprocally proportional, are equal to onethose angles equal which are opposite to the homologous sides. PROP. VII. THEOREM. If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals ; then, if each of the remaining angles be either... | 677.169 | 1 |
Trigonometry, a branch of mathematics that deals with the relationships between the angles and sides of triangles, is a fundamental concept in various fields such as physics, engineering, and architecture. One of the most important identities in trigonometry is the "cos a+b" identity, which allows us to express the cosine of the sum of two angles in terms of the cosines and sines of the individual angles. In this article, we will explore the significance of this identity, its applications in real-world scenarios, and how it can be derived and utilized effectively.
Understanding the "cos a+b" Identity
The "cos a+b" identity, also known as the cosine of a sum formula, states that:
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
This identity provides a way to calculate the cosine of the sum of two angles, given the cosines and sines of the individual angles. It is derived from the more general concept of the dot product between two vectors in a Cartesian coordinate system.
Deriving the "cos a+b" Identity
To understand the derivation of the "cos a+b" identity, let's consider two vectors, A and B, in a Cartesian coordinate system. The dot product of these vectors is given by:
A · B = |A||B|cos(θ)
where |A| and |B| represent the magnitudes of vectors A and B, respectively, and θ is the angle between them. By expressing vectors A and B in terms of their components, we can rewrite the dot product equation as:
A · B = (Axi + Ayj + Azk) · (Bxi + Byj + Bzk)
Expanding the dot product using the distributive property, we get:
A · B = AxBx + AyBy + AzBz
Now, let's consider two unit vectors, i and j, that are perpendicular to each other. The dot product of these vectors is:
i · j = |i||j|cos(90°) = 0
Since the dot product of perpendicular vectors is zero, we can rewrite the dot product equation as:
A · B = AxBx + AyBy + AzBz = |A||B|cos(θ)
Comparing the coefficients of the dot product equation and the "cos a+b" identity, we can equate the corresponding terms:
cos(θ) = cos(a + b)
AxBx = cos(a)cos(b)
AyBy = -sin(a)sin(b)
AzBz = 0
Therefore, we can conclude that:
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
Applications of the "cos a+b" Identity
The "cos a+b" identity finds numerous applications in various fields. Let's explore some of the practical scenarios where this identity is utilized:
1. Navigation and GPS Systems
In navigation and GPS systems, the "cos a+b" identity is used to calculate the position of an object based on its distance and direction from a reference point. By knowing the distances and angles between multiple reference points, the cosine of the sum of these angles can be calculated to determine the object's position accurately.
2. Engineering and Construction
In engineering and construction, the "cos a+b" identity is essential for calculating forces and moments acting on structures. By decomposing forces and moments into their components, engineers can use the "cos a+b" identity to determine the resultant forces and moments accurately.
3. Physics and Mechanics
In physics and mechanics, the "cos a+b" identity is used to analyze the motion of objects in two or three dimensions. By decomposing the motion into horizontal and vertical components, the "cos a+b" identity allows physicists to calculate the resultant motion accurately.
Examples and Case Studies
Let's explore a couple of examples and case studies to understand the practical applications of the "cos a+b" identity in more detail:
Example 1: Calculating the Resultant Force
Consider a scenario where two forces, F1 and F2, act on an object at angles θ1 and θ2 with respect to the horizontal axis, respectively. To calculate the resultant force, we can use the "cos a+b" identity as follows:
Resolve F1 and F2 into their horizontal and vertical components using trigonometry.
Calculate the horizontal and vertical components of the resultant force using the "cos a+b" identity.
Combine the horizontal and vertical components to obtain the magnitude and direction of the resultant force.
This example demonstrates how the "cos a+b" identity can be used to determine the resultant force accurately, considering the angles at which the individual forces act.
Case Study: Bridge Construction
In the construction of bridges, the "cos a+b" identity plays a crucial role in determining the forces acting on the bridge's components. Engineers use the identity to calculate the forces exerted by the weight of the bridge, wind loads, and other external factors. By considering the angles at which these forces act, the "cos a+b" identity allows engineers to design the bridge's structure to withstand the forces effectively.
Summary
The "cos a+b" identity is a powerful tool in trigonometry that allows us to calculate | 677.169 | 1 |
3-4-3-12 tiling
In geometry of the Euclidean plane, the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, and dodecagons, arranged in two vertex configuration: 3.4.3.12 and 3.12.12. The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling. There are 2 that contain both of these vertex figures among one more. It has square symmetry, p4m, [4,4], (*442). It is also called a demiregular tiling by some authors. (Wikipedia).
The theme is the tiling of flat plane by the hexagon which has the edges of 1,2,3,4,5,6 length, and that of other polygons of different edges. It is a very tough problem to make a tiling by a different edged polygon. Polygon tiling of plane often needs edges of the same lengths. It is well
In this mini-lecture, we explore tilings found in everyday life and give the mathematical definition of a tiling. In particular, we think about: (i) traditional Islamic tilings; (ii) floor, wallpaper, pavement, and architectural tilings; (iii) the three regular tilings using either equilat
This video is part of the #MegaFavNumbers project.
Domino tiling is a tessellation of the region in the Euclidean plane by dominos (2x1 rectangles).
In this video we consider square tilings. Sequence, where each element is equal to the number of tilings of an NxN square, is growing reall👉 Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties.
Given four points that repr
A complete walk through of OCR GCSE Maths June 2018 Higher Tier - Paper 5 non-calculator. Help revise for the J560 new specification 9-1 mathematics exams and your mock. This walkthrough tutorial has full solutions to each question, so you can use it like a mark scheme.
Choose to watc
These are the Mock Set (1) papers from Edexcel.
Mock Set (2) are all done (Higher ones), check them out
Pearson Education accepts no responsibility whatsoever for the accuracy or method of working in the answers given.
Click the PayPal Donate Link below to support my YouTube channel, i
ORGANIZERS : C. S. Aravinda and Rukmini Dey
DATE & TIME: 16 June 2018 to 25 June 2018
VENUE : Madhava Lecture Hall, ICTS, Bangalore
This workshop on geometry and topology for lecturers is aimed for participants who are lecturers in universities/institutes and colleges in India. This wi
Pearson Education accepts no responsibility whatsoever for the accuracy or method of working in the answers given.
Click the PayPal Donate Link below to support my YouTube channel, if you have found these videos helpful with your studies: | 677.169 | 1 |
Semicircle facts for kids
In geometry, a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.
Content is available under CC BY-SA 3.0 unless otherwise noted.
Kiddle encyclopedia articles are based on selected content and facts from Wikipedia, edited or rewritten for children.
Powered by MediaWiki. | 677.169 | 1 |
The first three books of Euclid's Elements of geometry, with theorems and ...
another and produced to meet the tangents drawn from the extremities of the bisecting line; the parts intercepted between the tangents and the circumferences are equal.
28. If two opposite angles of a trapezium be right angles, the angles subtended by either side at the two opposite angular points will be equal.
29. If a perpendicular be let fall from the right angle of a right-angled triangle on the hypotenuse, the rectangle contained by the hypotenuse, and either of the segments, into which it is divided by the perpendicular, is equal to the square of the side adjacent to that segment.
30. A quadrilateral figure may have a circle described about it, if the rectangle contained by the segments of the diagonal be equal.
31. If an equilateral triangle be inscribed in a circle, the square on a side thereof is equal to three times the square described upon the radius.
32. The vertical angle of an oblique-angled triangle inscribed in a circle, is greater or less than a right angle, by the angle contained by the base and the diameter drawn from the extremity of the base.
33. If a circle be inscribed in a right-angled triangle, the difference between the two sides containing the right angle and the hypotenuse, is equal to the diameter of the circle.
34. If a semicircle be inscribed in a right-angled triangle, so as to touch the hypotenuse and perpendicular, and from the extremity of its diameter a line to be drawn through the point of contact to meet the perpendicular produced; the part produced will be equal to the perpendicular.
GEOMETRICAL ANALYSIS.
In the method of analysis we assume the proposition advanced, and then proceed to trace the consequences which follow from this assumption, till we arrive at some known or admitted relation. The reverse of this process constitutes synthesis, or composition, which is the method employed in the preceding pages. In the solution of geometrical problems of more than ordinary difficulty, it is necessary that we should adopt the method of analysis, in order to discover the different steps which must be pursued in the construction. Analysis, observes an eminent geometer, presents the medium of invention; while synthesis naturally directs the course of instruction.
The following problem is given as an illustration of this method.
PROBLEM. Given the hypotenuse of a right-angled triangle, and the sum of the base and perpendicular, to construct the triangle.
D
A
ANALYSIS. Suppose the thing to be done. Let ABC be the triangle required, having BC equal to the given hypotenuse, the angle BAC equal to a right angle, and AB, AC together equal to the given sum of the base and perpendicular. On BA produced take AD equal to AC, and join CD.
B
C
Then BD is equal to the sum of AB and AC. And since ADC is a right-angled isosceles triangle, therefore the angles ADC and ACD are each of them half a right angle.
Hence we have the following composition:
SYNTHESIS. Make BD equal to the sum of AB and AC; from D draw (E. I. 11. and 9.) DC making the angle BDC equal to half a right angle; from B, the other extremity of BD, draw BC equal to the given hypotenuse, meeting DC in c; and from c (E. I. 12.) let fall the perpendicular CA; then ABC is the triangle required.
For the right-angled triangle ADC is evidently isosceles, and therefore AC is equal to AD, and the sum of AB and AC is equal to BD; wherefore, &c. | 677.169 | 1 |
PYTHAG - Review (calculations and applications)
Interactive practice questions
Which side of the triangle in the diagram is the hypotenuse?
A triangle with its sides labeled as $X$X, $Y$Y, and $Z$Z is shown. The vertex formed by the intersection of sides $X$X and $Y$Y is a right angle, denoted by a square symbol at the point of intersection.
$X$X
A
$Y$Y
B
$Z$Z
C
Easy
< 1min
Which side of the triangle in the diagram is the hypotenuse?
Easy
< 1min
We want to determine if the triple $\left(9,12,15\right)$(9,12,15) is a Pythagorean triple. | 677.169 | 1 |
Two lines that intersect and form right angles are called perpendicular lines. The symbol ⊥ is used to denote perpendicular lines. In Figure , line l ⊥ line m. Figure 2 Perpendicular lines.
Where do two perpendicular lines intersect point segment ray or line?
When 2 lines intersect at a right angle, they are perpendicular lines. We can also say that if 2 lines are perpendicular, then their intersection forms a right angle. Sometimes in everyday language, parts of lines (rays and line segments) that meet at right angles are also called perpendicular lines.
Where do two perpendicular lines intersect quizlet?
Two lines that intersect at right angles.
Is the image an example of perpendicular lines quizlet?
Is the image an example of perpendicular lines? Yes; the lines intersect at a right angle.
What are perpendicular lines?
Perpendicular lines are lines that intersect at a right (90 degrees) angle.
Do perpendicular lines have to intersect?
Perpendicular lines always intersect each other, however, all intersecting lines are not always perpendicular to each other. The two main properties of perpendicular lines are: Perpendicular lines always meet or intersect each other. The angle between any two perpendicular lines is always equal to 90.
Are two lines perpendicular if they intersect to form a right angle?
ex) perpendicular lines conditional If two lines are perpendicular, then they intersect to form right angles. converse If two lines intersect to form right angles, then they are perpendicular. … Biconditional Two lines intersect iff their intersection is exactly one point.
What happens when two perpendicular lines intersect?
A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles.
Which situation shows perpendicular lines?
Perpendicular lines occur anytime two lines meet at a 90° angle, also known as a right angle. Sometimes, you will see a little square in the corner of an angle, to show it is perpendicular. There are many examples of perpendicular lines in everyday life, including a football field and train tracks.
How do you find perpendicular lines?
Perpendicular lines intersect at right angles to one another. To figure out if two equations are perpendicular, take a look at their slopes. The slopes of perpendicular lines are opposite reciprocals of each other.
Where do perpendicular bisectors intersect?
The point of intersection of the perpendicular bisectors in a triangle is called its circumcenter. In an acute triangle, they meet inside a triangle, in an obtuse triangle they meet outside the triangle, and in right triangles, they meet at the hypotenuse.
How do you find the point of intersection of a perpendicular line?
To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.
Do two lines that are perpendicular intersect at exactly two points?
Two intersecting planes intersect in exactly one point. … In a plane, two lines perpendicular to the same line are parallel. In space, two lines perpendicular to the same line are parallel. If a line is perpendicular to a line in a plane, it is perpendicular to the plane.
Can two lines that are perpendicular to each other ever not intersect?
Two distinct lines intersecting each other at 90° or a right angle are called perpendicular lines. Here, AB is perpendicular to XY because AB and XY intersect each other at 90°. The two lines are parallel and do not intersect each other. They can never be perpendicular to each other.
What is the difference between perpendicular and intersecting lines?
What is the Difference Between Perpendicular and Intersecting Lines? When intersecting lines cross each other, there is no defined angle at which they meet, it can be any angle. However, perpendicular lines always intersect each other at right angles (90°).
Which best describes perpendicular lines?
In geometry, a branch of mathematics, perpendicular lines are defined as two lines that meet or intersect each other at right angles (90°).
Which of the following is the best example of perpendicular lines?
The way these two sets of lines cross makes them perpendicular lines. They are perpendicular because they cross at right angles to each other. Railroad tracks are a great example of natural perpendicular lines.
How many points do the rays intersect in an angle?
An angle is created when two rays connect at a common point. You can see that the two rays are connected at a common endpoint, called a vertex. This forms the angle. An angle is named by points on the rays.
What is an exact location in space?
Point. A point is an exact location in space. A point is denoted by a dot. A point has no size.
How do you know if two lines segments are perpendicular?
Answer: If two line segments intersect at 90 degrees, then two line segments are perpendicular. | 677.169 | 1 |
Can you please explain the Side-Side- Side Postulate, The Side-Angle-Side Postulate, and The Angle-Side-Angle Postulate.
Thank you very much
The side-side-side postulate is when two triangles are congruent with one another. (all sides are the same). The Side Angle Side postulate (SAS) states that if two side and the included angle of one trianle are congruent to two sides and the included angle of another triangle, then these two sides are congruent. The angle-side-angle postulate states that if two angles and the included side of on triangle are congruent to two angles and the included side of another triangle, then these two triangles are congruent.
To understand the Side-Side-Side (SSS) postulate, Side-Angle-Side (SAS) postulate, and Angle-Side-Angle (ASA) postulate, let's break them down one by one.
1. Side-Side-Side (SSS) Postulate:
The SSS postulate states that if the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. In other words, if all sides of two triangles are equal in length, they must be congruent.
To determine if two triangles satisfy the SSS postulate and are congruent, you need to compare the lengths of all three sides of both triangles. If all three sides of one triangle are equal to the corresponding sides of another triangle, then the triangles are congruent.
2. Side-Angle-Side (SAS) Postulate:
The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
To determine if two triangles satisfy the SAS postulate and are congruent, you need to compare the lengths of two sides and the angle between them of both triangles. If those measures are equal in both triangles, then the triangles are congruent.
3. Angle-Side-Angle (ASA) Postulate:
The ASA postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
To determine if two triangles satisfy the ASA postulate and are congruent, you need to compare the measures of two angles and the length of the side between those angles in both triangles. If those measures are equal in both triangles, then the triangles are congruent.
It's important to note that there are other postulates and theorems in geometry that can be used to prove triangle congruence. The SSS, SAS, and ASA postulates are just a few examples commonly used. | 677.169 | 1 |
We are learning to identify, measure, estimate and compare angles on a straight line, angle at a point and vertically opposite angles to find unknown angles.
This worksheet outlines the measurement of angles, including its unit, and how they are measured with a protractor.This is a part of a larger unit designed to help your students learn how to estimate, measure, draw and compare angles. | 677.169 | 1 |
The picture above was just taken at the National Museum of Mathematics in New York. As you can see, the tricycle has square wheels. I assure you the ride is smoothHere are some equations you may find helpful:
cosh-1(x) = ln(x +/- (x2-1)1/2)
The derivative of cosh-1(x) = 1/(x2-1)1/2
This one is worth a beer and an appetizer. However, the rule of 24-hour old on previous winners applies.
I spent a bit of time trying to derive a formula for the height of the curve above the floor. The dips in the curve are at height 0 (floor height). When the square is standing on its point with the point on the floor, the center of the square is sqrt(2) above the floor. The center must always be sqrt(2) above the floor for a smooth ride.
When the square is resting at the top of the curve, the center is located 1 above the top of the curve. That means the height of the curve above the floor is sqrt(2) - 1.
I messed around with triangles of the center, to the point of contact, and to the point on the ground sqrt(2) below the center, but didn't make much progress on a generalized formula. Either way, I doubt I would have been able to get the forumula in terms of cosh, even though that was in the hint you gave.
A quick google of "square wheels math" brought up a site at Vanderbilt that got me the formula I needed.
The professor used the same dimension you did to keep the math simple. The formula for the height of the curve is: y = sqrt(2) - cosh(x)
x = 0 is the maximum height of sqrt(2) - 1
To find the horizontal distance travelled, we need the distance to the minimum height of 0:
Tbh I have no idea where to even start. I mean if I thought about it for like a Long time I could maybe figure out the curvature of the partial circle. Then figure out what % of a circle that is. Then figure out the "sort of diameter" thing (it's not the real diameter because it doesn't go through the origin, or at least not necessarily). And for some reason my phone autocorrects the word Long to have a capital L. Wth?
So that's my detailed answer. Can someone at least confirm if the logic is sound?
Edit: Obviously you'd multiply the "sort of diameter" by 4, because that'd be one full revolutionThat was like the opposite of a hint. Now I'm even more confused. I haven't done calculus for over 6-7 years nowThere's more than one way to skin a cat. I can't see getting to the end without calculus though.
There's more than one way to skin a cat. I can't see getting to the end without calculus though.
I was wondering if you couldn't just use trig to solve, but am traveling and don't have easy access to paper and pen.
Start with the square wheel laying on a side flush on flat ground with one end on the origin line. As the wheel turns on a corner, the center of the square traces a quarter circle with radius of sqrt(2) and center (0,0) from (1,1) to (-1,1). Given the formula for that circle, we now know how we need to vary the height of the ground at any point to keep the center of the square at y=1. The trick though is that as you vary the height of the ground, you no longer trace a circle with the center, but instead pinch it along the x axis because it traces that shape along the arc length of the path rather than along the x axis.
The race is not always to the swift, nor the battle to the strong; but that is the way to betWizard, did you enjoy the math museum? I enjoy basic math and statistics but clearly not at your level. That museum is on my list the next time I get to New York.
I liked it but didn't love it. It seemed to be targeted to a very young age, like 4 to 12. There were lots of games and things to do, but I think most kids would not come away with the mathematical point behind whatever they were doing. To be honest, I didn't get the point behind a lot of the activities either. I probably would come away with a better review if I had more time, but as usual in New York, I spread myself rather thin and only had about 90 minutes there. The museum itself is rather small, I'd say about 3,000 square feet of public space. Many of the activities were difficult to understand how to control it or the object. There are lots of staff members floating about, mostly Asian males, probably college students, who were more than happy to discuss the math behind anything.
While my review is rather luke warm, I bought it up to my daughter's math teacher, who just raved about it. However, she went on a Christmas when nothing else in New York was open, so may have been grateful to do anythingIf "it" = the total tread of the wheel, then if the square wheel never slips, won't the total horizontal distance covered by the tread = 8?
Imagine a double exposure time lapse photo of the wheel, one exposure as it hits the peak of a bump, then another as it finishes a 1/4 turn, and the wheel reaches the peak of next bump. In the photo, would the edges of the square wheel overlap, align, or have a gap?
I imagine "align", which means if I took a four exposure time lapse photo, when the wheel hit the peak of each bump, wouldn't the wheel appear as four boxes next to each other, i.e., a rectangle with sides 2 and 8?
I took a look at the problem, but it came down to calculating curve lengths, which is a little outside of my expertise.
I do have one question: does the problem assume that, when the square is turned so that one of the vertices extends as far down as possible, it touches the point where two of the "bumps" of the curve meet? I have a feeling the problem's solution would be different if it did not.
I took a look at the problem, but it came down to calculating curve lengths, which is a little outside of my expertise.
It isn't easy to stump you.
Quote:
I do have one question: does the problem assume that, when the square is turned so that one of the vertices extends as far down as possible, it touches the point where two of the "bumps" of the curve meet? | 677.169 | 1 |
Planes A and B intersect at an angle. Intersection of lines is when two lines meets at a particular point and cuts each other at the same point. Its a measure of perpendicularity for right angles and greater or lesser for others.
At any point W, line m and line n cuts each other at point W to form an angle as shown from the diagram. | 677.169 | 1 |
STEP 1
Assumptions
1. Triangle ABCABCABC is given with ∠ABC=30∘\angle ABC = 30^{\circ}∠ABC=30∘.
2. The perpendicular bisector of side BCBCBC intersects ABABAB at MMM and ACACAC at DDD.
3. MDMDMD is perpendicular to BCBCBC and MD=2 cmMD = 2 \text{ cm}MD=2 cm.
4. We need to find the length of CMCMCM.
STEP 2
Since MDMDMD is the perpendicular bisector of BCBCBC, it means that BD=DCBD = DCBD=DC and MDMDMD is perpendicular to BCBCBC.
STEP 3
In △BMD\triangle BMD△BMD, since ∠BMD\angle BMD∠BMD is a right angle, we can use the definition of sine to find BDBDBD.
sin(∠ABC)=MDBD\sin(\angle ABC) = \frac{MD}{BD}sin(∠ABC)=BDMD
STEP 7
STEP 8
STEP 9
Now we can find the length of BMBMBM using the fact that △BMD\triangle BMD△BMD is a 30-60-90 right triangle, where the side opposite the 30-degree angle (BDBDBD) is half the length of the hypotenuse (BMBMBM).
STEP 10
Since BDBDBD is half the length of BMBMBM, we can write:
BM=2×BDBM = 2 \times BDBM=2×BD | 677.169 | 1 |
Fist all, we must aware What is Trigonometry ? It is a branch of mathematics that deals with the relationships between the angles and sides of triangles, as well as the functions derived from them. Trigonometry primarily focuses on the study of trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent, and their properties and applications. It has use in various fields including physics, engineering, astronomy, navigation, architecture, and more, where understanding and manipulating angles and distances are crucial.
The SSC CGL syllabus of Trigonometry includes questions from the Trigonometric ratios, Trigonometric identities, Heights and Distances, etc.
Topics Covered
Circular Measure of Angles (Radian & Degree Measure)
Trigonometric Ratios and Identities
Angle of Elevation
Angle of Depression
Miscellaneous Questions
Download : SSC CGL Trigonometry previous year questions pdf
Name : SSC Mathematics Trigonometry Previous Year Solved Paper Last 10 year Solved questions Medium : English Number of pages : 50 | 677.169 | 1 |
Ex 5.7, 2 (d) - Chapter 5 Class 6 Understanding Elementary Shapes
Last updated at April 16, 2024 by Teachoo
Transcript
Ex 5.7, 2 Give reasons for the following : (d) Squares, rectangles, parallelograms are all quadrilaterals. Since square, rectangle, parallelogram have 4 sides.
They are all quadrilaterals.
∴ Square, Rectangle, Parallelogram are all quadrilaterals | 677.169 | 1 |
1a) (i) Parallax Error: This occurs when the line of sight is not perpendicular to the scale of the measuring instrument. It can lead to a false reading if not properly corrected.
(ii) Instrumental Error: This error is associated with the precision and calibration of the measuring instrument. If the instrument is not calibrated correctly or if it has inherent inaccuracies, it can introduce errors into the measurements.
(iii) Human Error: Mistakes made by the person taking measurements can lead to errors. This can include misreading the scale, not aligning the instrument properly, or inaccurately recording the measurement.
(1b) When considering a line on a graph, the slope is defined as the change in the vertical direction divided by the change in the horizontal direction. This ratio represents how much the dependent variable (usually the y-axis) changes for every unit change in the independent variable (usually the x-axis).
In terms of distance, if we have a sloped line on a map or a coordinate plane, its slope can impact the actual distance between two points. The steeper the slope, the greater the distance covered between the two points. Conversely, a flatter slope corresponds to a smaller distance.
For height measurements, the slope can determine the elevation change between two points. A steep slope indicates a significant change in height over a short distance, while a gentle slope suggests a gradual change in elevation.
In the case of length or width measurements, the slope can affect the dimensions. For example, if you measure the length of a ramp with a certain slope, a steeper slope will result in a longer measured distance, while a shallower slope will yield a shorter measurement.
It's important to note that the units of measurement must be consistent for accurate interpretations of slope and linear measurements. Additionally, in real-world scenarios, other factors such as topography, terrain, and surface characteristics can also influence linear measurements.
In summary, the slope of a line can impact linear measurements by affecting the distances, heights, lengths, and elevations between points2) (i) Survey the site: Prior to setting out the heading, conduct a survey of the site to gather relevant information. This may include understanding the site plan, examining any existing structures or utilities around the excavation area, and identifying any potential hazards.
(ii) Determine alignment and dimensions: Based on the required alignment and dimensions specified in the construction plans or regulations, mark the starting and ending points of the trench. This can be done using pegs, stakes, or other markers.
(iii) Strap the reference line: Establish a reference line, also known as a baseline, along the proposed direction of the trench. This line acts as a guide for excavation activities. Secure a string or tape tightly between two sturdy reference points, such as two pegs, located outside the trench boundaries.
(iv) Measure and mark offsets: Use measuring tapes or total stations to mark the required offsets from the reference line. These offsets represent the dimensions of the trench, such as width, depth, or any other specifications. Mark these points along the reference line at regular intervals.
(v) Set up batter boards: Install batter boards at the corners of the trench to provide a stable reference point for the excavation. Batter boards are temporary structures that have plumb lines attached to indicate the exact position of the trench boundaries.
(vi) Verify alignment and levels: Use a level or a laser level to ensure that the batter boards and the trench alignment are vertical and in the correct position. Adjust the batter boards as necessary to achieve the desired alignment and levels.
(vii) Transfer heading marks to the ground: From the marked reference points and offsets along the reference line, transfer the measurements onto the ground using spray paint, chalk, or other suitable marking tools. This will create a clear outline of the trench boundaries.
(viii) Cross-check and double-check: After marking the trench boundaries on the ground, cross-check the alignment and dimensions against the construction plans or specifications to ensure accuracy. Make any adjustments if needed.
(ix) Begin excavation: With the heading set out, excavation can commence along the marked trench boundaries. Continue to monitor the alignment and dimensions throughout the excavation processii) Moisture Control: The cavity allows for better drainage and ventilation, reducing the risk of moisture penetration and dampness. This contributes to the longevity of the wall and helps prevent issues such as mold growth.
(iii) Sound Insulation: Cavity walls can offer improved sound insulation compared to solid walls. The air gap acts as a buffer, reducing the transmission of sound from one side of the wall to the other.
(iv) Structural Stability: Cavity walls often provide better structural stability and resistance to cracking. The separation of the inner and outer leaves can help distribute loads more effectively.
(v) Flexibility in Construction: Cavity walls allow for flexibility in incorporating different materials in the inner and outer leaves, optimizing the wall for both structural and insulation purposes4)5) (i) Load: The type and magnitude of the load that the formwork will be subjected to, such as the weight of concrete, construction equipment, and workers, will determine the strength and stability requirements of the formwork design.
(ii) Concrete mix: The properties of the concrete mix, including its consistency, workability, and curing time, will influence the design of formwork. The formwork needs to be able to support the weight of the concrete and maintain its shape until it sets.
(iii) Construction schedule: The time available for formwork assembly, concrete pouring, and formwork removal affects the design. Fast-paced projects may require formwork systems that can be easily assembled, disassembled, and reused, while longer construction schedules may allow for more traditional formwork methods.
(iv) Architectural requirements: The desired shape, size, and finish of the concrete structure will impact the formwork design. Complex architectural designs may necessitate custom formwork systems or the use of special formwork materials to achieve the desired resulti) Surface preparation: Begin by prepping the wall surface. Remove any loose paint, dirt, or debris. Repair any cracks or holes with an appropriate filler and sand the surface to ensure it is smooth and even.
(ii) Primer application: Apply a coat of primer to the wall. This helps to seal the surface and provides a good base for the rendering material to adhere to.
(iii) Mixing the rendering material: Depending on the type of smooth finish you want to achieve, prepare the rendering material accordingly. Common options include plaster, joint compound, or specialized smoothing compounds.
(iv) Application of the render: Start by applying a thin layer of the rendering material onto the wall using a trowel or a similar tool. Spread it evenly across the surface, taking care to avoid any air bubbles or uneven patches.
(v) Smoothing the surface: Once the initial layer is applied, use a trowel, spatula, or a smoothing tool to flatten and smooth the rendering material. Work in small sections, ensuring an even and consistent surface.
(vi) Drying and curing: Allow the first layer to dry completely as per the manufacturer's instructions. Typically, this can take a few hours to a day depending on the rendering material used.
(vii) Additional layers and sanding: Depending on the desired smoothness, you may need to apply additional layers of the rendering material, repeating the smoothing process between each layer. After the final layer, sand the surface lightly to achieve a perfectly smooth finish.
(7ii) SCRAPED FINISH:
(i) Surface preparation: Prepare the wall surface by cleaning it thoroughly, removing any loose paint, dirt, or debris. Repair any cracks or holes and ensure the surface is even and smooth.
(ii) Primer application: Apply a coat of primer on the wall to seal the surface and provide a good base for the rendering material.
(iii) Mixing the rendering material: Prepare the rendering material according to the manufacturer's instructions. Typically, for a scraped finish, a textured compound or stucco mix is used.
(iv) Application of the render: Apply the rendering material to the wall using a trowel or a similar tool. Spread an even layer across the surface, leaving the desired texture. You can manipulate the texture by applying different techniques like swirls, lines, or patterns.
(v) Scraping: Once the rendering material starts to set but is still slightly wet, use a scraping tool (such as a trowel or scraper) to create the desired finish. Gently scrape the surface in a consistent motion, removing excess material and revealing the textured pattern.
(vi) Cleaning and finishing: Clean any excess material from the surrounding areas and smooth out any imperfections. Allow the rendering material to dry completely as per the manufacturer's instructions.
(vii) Optional sealing and painting: Depending on the material used, you may choose to seal the scraped finish to protect it from moisture. Additionally, you can paint over the scraped finish to enhance its appearance and durabilityiv) Insulating materials should be resistant to chemical degradation and environmental factors10) (i) Bituminous Materials: Bituminous materials, such as bitumen or asphalt-based products, are commonly used for damp-proofing. Bituminous coatings and membranes are applied to foundations and walls to create a barrier that prevents the penetration of moisture.
(ii) Polyethylene Sheeting: Polyethylene sheeting, often in the form of plastic membranes, is used as a damp-proofing material. These sheets are applied to foundation walls to create a barrier against water vapor and moisture infiltration. They are especially effective in preventing moisture from the ground.
(iii) Cementitious Coatings: Cementitious damp-proofing coatings are formulated with cement-based materials that create a waterproof barrier when applied to surfaces. These coatings can be used on foundations and walls to protect against the ingress of moisture.
(iv) Liquid Membranes: Liquid membrane damp-proofing materials are applied as a liquid that forms a continuous, seamless membrane when cured. These materials are often polymer-based and provide effective protection against water penetration. Liquid membranes can be sprayed or applied as a brush-on coatingSECTION B: ANSWER FIVE(5) QUESTIONS ONLY
(11a) (i) Oversite Concrete: Oversite concrete refers to the layer of concrete that is laid over the compacted **** or other suitable sub-base to provide a smooth and even surface for constructing the floor of a building. It serves as a foundation for the flooring material (e.g., tiles, wood, or carpet) and helps to create a level and stable base for the structure.
(ii) Datum Level: Datum level, also known as the reference level or benchmark, is a fixed point or surface used as a base elevation from which other levels and measurements are taken. It serves as a reference point for establishing the heights or depths of various elements in a construction project. Datum levels are crucial for ensuring consistency and accuracy in vertical measurements.
(iii) Plumbing: In the context of construction, plumbing refers to the alignment or positioning of a structure, component, or element in a vertical and straight orientation. It ensures that walls, columns, or other structural elements are upright and not leaning or tilting. Plumbed structures contribute to the overall stability and aesthetics of a building.
(iv) Leveling: Leveling is a surveying and construction technique used to determine the relative heights or elevations of different points on the Earth's surface. It involves the use of leveling instruments, such as a level or a theodolite, to establish horizontal lines and measure vertical distances. Leveling is essential for creating level surfaces, aligning structures, and ensuring the proper drainage of fluids.
(11b) (i) Durability: Steel measuring chains are generally more durable than metallic linen tapes. Steel is resistant to wear, tear, and corrosion, making it suitable for use in various weather conditions and challenging environments. Linen tapes may wear out more quickly, especially in harsh conditions.
(ii) Accuracy: Steel measuring chains tend to provide more accurate measurements compared to metallic linen tapes. Steel has less stretch or deformation under tension, resulting in more reliable and consistent measurements. Linen tapes may stretch over time, leading to inaccuracies in measurements.
(iii) Longevity: Steel measuring chains have a longer lifespan than metallic linen tapes. They are less prone to damage and degradation, ensuring that they can be used for an extended period without a significant loss of accuracy. Linen tapes may be more susceptible to damage, affecting their longevity.
(iv) Versatility: Steel measuring chains are versatile and can be used in a variety of surveying and measurement applications. They are suitable for both short and long-distance measurements. Linen tapes may have limitations in terms of length and may not be as versatile in certain situations12a) Draw the diagram
(12b) Draw the diagram
(12c) Draw the diagram13a) (i) Strength and Stability: Formwork must be structurally sound and possess the strength and stability to support the weight of fresh concrete and any additional loads, such as construction workers and equipment.
(iii) Waterproofing: Formwork should be resistant to water to prevent the absorption of moisture from fresh concrete, which can affect the curing process and lead to surface defects.
(iv) Ease of Installation and Removal: Formwork systems should be designed for easy and efficient installation as well as removal.
(13bi) CEMENT: (i) Setting time test: This test measures the time it takes for cement to set and harden. It helps determine if the cement meets the required setting time specifications.
(ii) Consistency test: This test assesses the consistency or workability of the cement paste. It involves measuring the amount of water required to achieve a standard consistency, which can indicate the quality of the cement.
(13bii) SAND: (i) Sieve analysis: This test involves passing the sand through a series of sieves with different mesh sizes. It helps determine the particle size distribution of the sand, which can affect its suitability for various applications.
(ii) Moisture content test: This test measures the amount of moisture present in the sand. It is important to ensure that the sand is neither too wet nor too dry, as this can affect the workability and strength of the concrete14a) In reinforced concrete, the steel reinforcement is passive and only begins to resist tensile forces once cracking occurs. In contrast, prestressed concrete introduces internal stresses in the reinforcement prior to loading, allowing it to actively counteract the tensile forces.
(14b) (i) Pre-tensioning: Pre-tensioning is a method of prestressing concrete where the steel reinforcement, typically in the form of high-strength steel wires or strands, is tensioned before placing the concrete. The tension in the reinforcement is achieved by stressing the steel wires or strands using hydraulic jacks or other suitable mechanisms. Once the steel is tensioned, it is anchored to the formwork, and concrete is cast around it. As the concrete sets and hardens, it firmly bonds with the tensioned steel reinforcement. After the concrete gains sufficient strength, the tension is released from the steel, which enables it to transfer the compressive forces to the concrete, resulting in a prestressed structure.
(ii) Post-tensioning: Post-tensioning is another method of prestressing concrete, but it differs from pre-tensioning in terms of when the steel reinforcement is tensioned. In post-tensioning, the steel tendons or cables are not tensioned until after the concrete has hardened. Ducts or sleeves are provided in the concrete during the casting, and the tendons are passed through these ducts. Once the concrete has sufficiently hardened, the tendons are tensioned using hydraulic jacks. The tension is then locked in place by anchorages, and the tendons transfer the compressive forces to the hardened concrete. Post-tensioning allows for flexibility in adjusting the prestressing force, providing a more versatile method for accommodating variations in construction and design requirements17ai) PAINT:
(i) Pigment: Paint contains pigments, which are finely ground particles that provide color and opacity. They determine the visual appearance of the paint once it dries.
(ii) Binder: The binder acts as the film-forming component in paint. It holds the pigment particles together and allows them to adhere to the surface. Common binders include acrylic, latex, alkyd, oil, or epoxy.
(iii) Vehicle: The vehicle is the liquid portion of paint that carries the pigment and binder. It provides the paint with the desired consistency for application. The choice of vehicle affects the drying time, durability, and ease of application of the paint.
(iv) Additives: Paint can contain various additives to enhance its properties, such as thickeners to adjust viscosity, drying agents to speed up drying, anti-settling agents to prevent pigment settling, and UV inhibitors to protect against fading.
(v) Durability: Paint should provide a durable and long-lasting finish. Its resistance to wear, weathering, and chemicals depends on the type of binder used, additives incorporated, and the application method.
(vi) Coverage: Paint should offer good coverage to hide the underlying surface and provide an even color. The opacity of paint is determined by the composition and concentration of pigments.
(vii) Gloss/sheen: Paint can have different levels of sheen, ranging from flat (no sheen) to high gloss. The sheen affects the appearance of the painted surface and can also influence durability and cleanability.
(viii) Application: Paint can be applied using various techniques such as brushing, rolling, or spraying. The consistency of the paint, as well as the application method, affects the evenness and quality of the finish.
(17aii) VARNISH:
(i) Protection: Varnish forms a durable protective layer over the surface it is applied to, shielding it from scratches, moisture, UV radiation, and general wear and tear. It helps extend the lifespan of the material.
(ii) Gloss and Clarity: Varnish can provide a glossy finish that enhances the natural beauty of the surface. It can also be available in different levels of sheen, allowing for a desired aesthetic appearance.
(iii) Transparency: Varnish is usually transparent or semi-transparent, allowing the underlying surface to show through. This property is particularly desirable when preserving the natural grain or color of wood.
(iv) Hardness: Varnish should dry to a hard and durable finish to withstand impacts and resist scratches. It provides a protective barrier that can withstand repeated use and cleaning.
(v) Drying Time: Varnish has a specific drying time, during which it becomes tack-free and cures fully to form a hard surface. The drying time varies based on the type of varnish and environmental conditions.
(vi) Application: Varnish can be applied using various methods such as brushing, spraying, or dipping. It is important to ensure proper application techniques to achieve an even and smooth finish.
(vii) Compatibility: Varnish should be compatible with the material it is applied to. Different varnishes are formulated for specific surfaces such as wood, metal, or concrete. Using the appropriate varnish ensures proper adhesion and compatibility with the substrate.
(viii) Maintenance: Varnished surfaces may require periodic maintenance to retain their appearance and protective properties. This can involve cleaning, reapplication, or refinishing, depending on the specific varnish and conditions of use.
(17bi) CRAZING:
(i) Incompatibility of Coatings: Crazing can occur when incompatible coatings are applied over each other. For example, applying a new coat of paint that is not compatible with the existing paint or primer may lead to crazing.
(ii) Excessive Thickness: Applying paint in excessively thick layers can result in crazing. As the paint dries, the outer layer may dry faster than the inner layers, causing stress and leading to the formation of fine cracks.
(17bii) PEELING:
(i) Poor Surface Preparation: Inadequate surface preparation, such as failure to remove dirt, grease, or loose old paint before applying a new coat, can lead to poor adhesion. This can result in peeling as the new paint layer does not bond properly with the substrate.
(ii) Moisture Infiltration: Moisture infiltration into the substrate, especially wood, can cause peeling. Moisture may lead to the breakdown of the bond between the paint and the surface, resulting in the paint layer detaching over time. Proper moisture protection and sealing are essential to prevent this failure | 677.169 | 1 |
Surface Properties of a Kite – Definition With Examples
Welcome to the fascinating world of geometry through the lens of Brighterly, where we make math fun and engaging for children! Today, we are going to explore the surface properties of a kite, a geometric shape that holds a special place in our hearts. A kite is not just a delightful toy that dances in the wind; it's a treasure trove of mathematical properties waiting to be discovered. With its unique angles, sides, and diagonals, a kite's geometry is a vibrant blend of art and science. Join us on this exhilarating journey as we delve into the definition, properties, and real-world examples of a kite. Through interactive exercises, lively explanations, and hands-on activities, we'll unlock the secrets of this extraordinary shape. So, grab your math caps and let your imagination soar with Brighterly!
What Are Surface Properties of a Kite?
The surface properties of a kite encompass the unique characteristics and attributes related to the angles, sides, and diagonals of a geometric shape known as a kite. Understanding these properties enables a deeper appreciation of geometry and its applications in everyday life. With a colorful blend of mathematical rigor and engaging examples, we'll unravel these properties to make them as fun and accessible as flying a kite on a breezy day!
Definition of a Kite in Geometry
A kite in geometry is a special type of quadrilateral, a figure with four sides. It consists of two pairs of adjacent sides that are equal in length. In other words, a kite is formed when you take two sticks of the same length and attach them together twice, creating a fluttering shape. Unlike a rhombus or a square, the angles and sides in a kite do not have to be equal, giving it a unique and interesting form. Let's delve into these intriguing surface properties!
Surface Properties of a Kite
The surface properties of a kite encompass the angles, sides, and diagonals that make up this unique shape. These properties hold the key to understanding the kite's geometry and provide the foundation for solving problems and drawing connections with other quadrilaterals. Whether it's the symmetry of the diagonals or the distinct lengths of the sides, the surface properties provide a mathematical playground for exploration.
Properties of Angles in a Kite
Angles in a kite are fascinating! In a kite, the angles formed by the equal-length sides are typically unequal, creating an asymmetrical appearance. The angles between the two shorter equal sides and the two longer equal sides have different measurements. Additionally, the diagonals intersect at a right angle, further contributing to the kite's unique appearance. By understanding these angles, one can design creative kite shapes for fun or study their behavior in mathematical applications.
Properties of Sides in a Kite
The sides in a kite are intriguing in their own right. While a kite does have two pairs of adjacent equal-length sides, these pairs are not necessarily the same length as each other. This difference creates a dynamic shape, giving the kite its distinct look. Whether you're drawing a kite for artistic purposes or calculating its area, the properties of its sides will help you achieve your goal with precision and flair.
Properties of Diagonals in a Kite
The diagonals of a kite are equally captivating. They intersect at a right angle, and one of the diagonals is bisected by the other. This unique feature separates the kite from other quadrilaterals and adds complexity to the shape. It's like a mathematical dance where the diagonals embrace and twirl, adding an elegant touch to the kite's structure.
Understanding the Angles in a Kite
Angles in a kite are not just abstract concepts; they play a practical role in design and problem-solving. Whether you are crafting a physical kite or exploring geometric patterns, understanding these angles provides a gateway to creativity and mathematical depth. Their asymmetry and unique relationships bring a burst of excitement to geometry, turning a simple shape into a rich field of exploration.
Examples of Angles in a Kite
Imagine designing a kite with one angle measuring 60 degrees and the other 120 degrees. How would that affect its flight? Or consider the angles in the logo of a famous brand. By examining these real-world examples, we can see how the angles in a kite translate into practical applications, enhancing our understanding and appreciation of this versatile shape.
Understanding the Sides in a Kite
The sides in a kite are more than lines on paper; they form the skeleton of the shape. By altering the length of the sides, you can create various kite forms, each with its personality and characteristics. From architectural designs to puzzles, the sides in a kite enable creativity and precision, turning simple lines into artistic expressions and mathematical marvels.
Examples of Sides in a Kite
Think of a kite-shaped window in a modern building or the wings of certain aircraft. The sides of these real-life kites impact their functionality and aesthetics. By exploring these examples, we gain insights into how the properties of the sides in a kite play a vital role in design, engineering, and even entertainment.
Understanding the Diagonals in a Kite
The diagonals in a kite are like the hidden threads that bind the shape together. Intersecting at right angles and with one diagonal bisected, they offer a glimpse into the symmetrical beauty and complexity of the kite. Their interaction is a dance of geometry, weaving together form and function in a mesmerizing pattern.
Examples of Diagonals in a Kite
From the blueprints of innovative buildings to the intricate designs of jewelry, examples of diagonals in a kite are everywhere. Their properties not only add aesthetic value but also contribute to structural integrity and design efficiency. Exploring these examples opens up new horizons in art, architecture, and more.
Difference Between a Kite and Other Quadrilaterals
While a kite shares some similarities with other quadrilaterals like parallelograms or trapezoids, it stands apart in its angles, sides, and diagonals. Unlike other quadrilaterals, a kite's diagonals intersect at right angles, and its sides form two distinct pairs of equal lengths. This sets it apart, creating a special niche within the family of quadrilaterals.
Practice Problems on Surface Properties of a Kite
Calculate the area of a kite with sides of 5 cm and 7 cm and one diagonal of 6 cm.
Design a kite shape that incorporates specific angles and side lengths for a school project.
Identify real-world examples of kites in architecture or nature, and analyze their properties.
Conclusion
Thank you for exploring the surface properties of a kite with Brighterly! We've embarked on a mathematical adventure, unraveling the mysteries of angles, sides, and diagonals in a kite. We've learned how these properties translate into real-world applications and inspire creativity in art, architecture, and even entertainment. Here at Brighterly, we believe that mathematics is a gateway to endless imagination and discovery. With each shape and figure, there's a story to tell, a problem to solve, and a world to explore. Keep playing, questioning, and learning with Brighterly, where every child is a mathematician, every shape is an adventure, and every problem is a chance to grow. See you in our next exciting exploration!
Frequently Asked Questions on Surface Properties of a Kite
What are the main characteristics of a kite?
A kite in geometry has two pairs of adjacent sides of equal length, and its diagonals intersect at right angles. Unlike other quadrilaterals, the angles formed by these sides can be asymmetrical. These distinct properties set the kite apart, making it a versatile shape in various applications. Brighterly's interactive lessons can help you explore these characteristics further.
Can I design my own kite using these properties?
Absolutely! Understanding the angles, sides, and diagonals of a kite allows you to craft unique and functional kites. Whether it's for a school project or a weekend hobby, the geometric principles of a kite can be applied in design and engineering. With Brighterly's hands-on activities, you can create your own kite, experimenting with different shapes, sizes, and aesthetics.
How does the kite differ from other quadrilaterals?
The kite is distinct from other quadrilaterals like parallelograms or trapezoids in its surface properties. While some quadrilaterals may have parallel sides or equal angles, the kite's diagonals intersect at right angles, and its sides form two different pairs of equal lengths. These unique properties give the kite its special place within the family of quadrilaterals. Brighterly's engaging lessons offer a deeper understanding of these differences, fostering a love for geometryBrackets in Math – Definition with Examples
Welcome to Brighterly's fun-filled world of mathematics, where we unravel the mysteries of numbers and equations to make learning a joyous journey. Today, we're exploring a fascinating aspect of math – brackets. These seemingly simple symbols – parentheses (), curly brackets {}, and square brackets [] – are silent powerhouses, steering the course of calculations […]
How to Find the Rank of the Matrix?
Welcome to Brighterly, where we make learning math exciting and accessible for kids! In this article, we'll explore the concept of matrix rank, an essential topic in linear algebra. Understanding the rank of a matrix is not only crucial for higher mathematics but also forms the foundation for various applications in science and engineering. At […]
Rounding Decimals – Definition with Examples
Welcome to another exciting topic on Brighterly, where we make learning math fun and engaging for children. Today, we're diving into the world of rounding decimals. This is a fundamental concept in mathematics that helps simplify complex numbers for easier understanding and calculation. Rounding decimals involves reducing the number of digits in a decimal while | 677.169 | 1 |
88.
УелЯдб 81 ... rectangle contained by the two straight lines , is equal to the rectangles contained by the several parts of the one and the several parts of the other . Let the given straight line AB be divided into E A I B N H K M C G D G any parts ...
УелЯдб 82 ... rectangle AN is equal to the rectangles contained by AE and CG , by EF and CG , and by FB and CG ; also the rectangle IK is equal to the rectangles contained by HL and GD , by LM and GD , and by MK and GD ; but ( E. xxxiv . 1. ) HL = AE ...
УелЯдб 83 ... rectangle contained by AB and CD is equal to the square of AE taken as often as is indicated by the product of the ... contained by the two parts , which are the greatest and the least , is less than the rectangle contained by the other ...
УелЯдб 84 ... triangle , if a straight line be drawn from the vertex to any point in the base , the square upon this line , together with the rectangle contained by the segments of the base , is equal to the square upon either of the equal sides ...
УелЯдб 85 ... rectangle contained by the aggregate and the difference of two unequal straight lines is equal to the difference of their squares . Let AC and CB be two unequal straight lines , A C D B of which CB is the greater ; and let them be | 677.169 | 1 |
Measuring Angles Art Idea
I love integrating art and maths! It is an awesome way to bring a difficult or more boring Maths topic to life.
When you are teaching your students how to measure angles to your upper primary students, art is the most fun way to get your kiddos excited about the topic.
Today I thought I would share with you one of my most favourite Maths + Art ideas!! The result is truly amazing and it ticks HEAPS of teaching and learning goals at the same time.
Curriculum links
But before I share with you what the activity is, let me explain what outcomes this unit can be ticking off (or at the very least contributing to the learning of!): Year 6 Maths Identify the relationships between angles on a straight line, angles at a point and vertically opposite angles; use these to determine unknown angles, communicating reasoning (AC9M6M04) Year 6 5 Maths Estimate, construct and measure angles in degrees, using appropriate tools including a protractor, and relate these measures to angle names (AC9M5M04) Year 5 4 Maths estimate and compare angles using angle names including acute, obtuse, straight angle, reflex and revolution, and recognise their relationship to a right angle (AC9M4M04) Year 4 Visual Arts Experiment with a range of ways to use visual conventions, visual arts processes and materials (AC9AVA4D01). Use visual conventions, visual arts processes and materials to create artworks that communicate ideas, perspectives and/or meaning (AC9AVA4C01).
I probably wouldn't try this activity with kiddos any younger than year 4. I'm not sure they would completely be able to visualise and implement the skills required to complete the artwork. The exploration of angles below year 3 is a lot simpler too, so other activities may be of more assistance with their learning at this age level.
The Process
Step 1 Trace a straight-edged outline of your animal onto white paper. Use a window or light box to help you see the image clearly.
Step 2 Place 3-5 dots within your image to become points of interest.
Step 3 Use a ruler and pencil to draw lines from the points of interests to the intersections and the outline of the body to create angles within your image.
Step 4 Go over your pencil lines with a fine liner and erase the pencil markings.
Step 5 Have your students then colour in each section they have created inside their animal outline. You can also then have the cut out their animal and glue it next to the original image onto a black piece of card.
Step 6 Have your students try to find all the different types of angles they have learnt about.
What do you reckon? Will you try this activity next time you are teaching angles to your upper primary students?In Year 6 science, we have a unit called Reef Warriors. The children explore how the growth and survival of living things are affected by the physical conditions of their environment. What better way to explore this concept than by looking at the Great Barrier Reef ecosystem! One that is in a constant battle of balance so that the plethora of species that call it home can continue to thrive and survive.
As part of this unit we encourage teachers to recreate the reef with their students using this fun Coral Reef Sculpture Art Idea.
And I thought it was just so fun that I couldn't just keep it hidden in the unit. It had to be shared! So here it is… the steps we followed to create a Coral Reef Masterpiece!
What You Will Need:
Carboard from boxes
Alfoil
Cardstock
Collage paper
Recycled materials
Glue
Scissors
Crayola Model Magic clay
Hot glue gun (optional)
Instructions:
Step 1: To begin this project I cut out the sculpture bases 20cmx10cm (but you can use any sized base really – just remember though, whatever size you do choose to use, the larger the space your students will need to cover with their reef – so you probably don't want your base to be too large!) and covered them with alfoil (you can get your students to do this part if you like).
Step 2: Then, after showing your students an example of one you have made up prior to class (which I find is best practice when doing art projects to help students see the bigger picture before they begin) have your students use the collage paper and cardstock to experiment with different folding techniques and arrangements on their alfoil base to come up with reef looking structures. Your students may like to try rolling their paper, folding their paper accordion style, and cutting their paper into a fringe and then rolling it. These techniques all work great from my experience. Maybe have them show you their structures before they glue them down but if you don't have time for this encourage your students to at least place their pieces where they would like them to go before gluing so they can plan their layout.
Step 3: Allow your students to glue their paper structures to the base.
Step 4: Once they have these paper structures in place, your students can now begin using the Model Magic clay (this clay will air dry so is fantastic for simple projects like this). The clay can be used to create additional coral pieces or be used to help hold other things in place like their recycled materials (bottle caps, pipecleaners, straws, etc) that they would like to add to their reef to create different looking coral structures. If your students are feeling creative they may even like to add some animals living among their coral using the materials available to them.
And there you have it! A super simple art idea, that is delightfully open-ended and will result in not only some fantastic art sculptures but also some pretty happy students as well!
At the beginning of the year, within the first week of school, I like to get something up in the classroom that personalises the classroom for the new cohort of students I am teaching. This usually takes the form of some sort of art, whether it be a self-portrait, name art or some sort of personal goal pop art exercise, these little touches from the students themselves really bring the classroom to life!
So today I thought I would share with you a fun name art exercise which as always can be modified to suit any age level. The way I have designed this particular exercise to share with you today, will allow your students to explore mixing primary colours to make new colours. And for the lower primary grades, it also gives your students a fun way to practise writing their name.
So to get started you will need: – table salt – Edicol dyes (blue, red, yellow) – paintbrushes/pipettes (I prefer paintbrushes for this as the amount of paint added can be better controlled for littlies with a paintbrush but up to you) – craft glue (in squeezy bottles) – A3/A4 cartridge paper (this paper is thicker than normal printer A3 paper, therefore withstanding higher volumes of liquid without tearing) – lead pencils – plastic or baking tray
Here's how to make your very own salt name art: 1. Give each student an A3/A4 piece of paper. Have them write their name on the piece of paper using their lead pencil and then go over the pencil lines with craft glue – demonstrate squeezing the glue to follow the lines before letting your students loose with the glue!
I used Aquadhere because that is all I had but any child friendly craft glue will work fine.
2. After their name has been fully written with glue, give students some salt to shake over their paper (students will need a lot of salt to ensure it is all covered). Demonstrate to students how to gently lift the paper and tip any excess into a tray at their table (alternatively a teacher or teacher aide may like to assist younger students).
Shake on salt – ensuring coverage of the whole name.
Have a tray on hand for students to shake off excess into.
3. Once the excess salt has been removed, introduce your students to the Edicol dyes. Demonstrate the following to the students: dip just the tip of your paintbrush into the dye (they won't need a lot of paint to make this work). Wipe off any excess on the sides of the dye container. Touch the salt gently with the paintbrush tip. The dye will transfer to the salt and the salt will spread the colour along the letter.
Encourage students to touch the salt gently – they don't need to wipe the brush along, the salt will carry the colour along the letter.
4. Using the same technique as described above, encourage your students to use more of the colours to cover the letters of their name. What do they observe? The colours will mix with the other colours, creating new colours. This should prompt a great discussion about mixing primary colours.
Finished product!
Annd you are done! Well done!! I hope you and your students enjoy decorating your classroom with these masterpieces this year. Don't forget you can send me photos to share on social media either by tagging me in your Instagram posts @ridgydidgeresources or by sending them to my email: [email protected] | 677.169 | 1 |
Segment Bisector Definition
How it works ?
In the applet below, the yellow line is said to be a BISECTOR (or SEGMENT BISECTOR) of the purple segment (with white endpoints.)
Interact with this applet for a minute, then answer the questions that follow.
Be sure to change the locations of the both white points and the yellow point each time before re-sliding the slider!
1.
How would you describe point C(with respect to the entire purple segment with white endpoints?)
Type your answer here…
2.
Complete the following sentence definition:
A SEGMENT BISECTOR is a _________________ that intersects a segment at its ______________ | 677.169 | 1 |
Man did I have you pegged.It's always the same with you guys. I asked you to define that in terms that are useful to me. You have refused to do so and have refused to admit you don't understand. Why is that?
There is no better definition that is more useful to you, other than your own definition, and you have refused to provide thisThere is no better definition that is more useful to you, other than your own definition, and you have refused to provide this.This is not what I asked you. I asked you, "If you disagree with his statement, then please show how the perimeter changes after each step." You once again try to take a circular path with a time element and turn it into a circle with no time element.
Gravock
LOL, as has been pointed out to you as well as Miles Mathis, the method fails to correctly determine the length of even a single line segment that is not parallel to one of the ordinate axes. The perimeter of the object doesn't change: It is what it is. And the hapless Mathis method of determining that perimeter starts with an inaccurate estimate and never improvesWhy are you trying to change the subject by bringing in Pythagorean theorem? The Pythagorean theorem does not hold in a non-Euclidean geometry. Google it!It is you, who is high. Pi is not my concept and neither is a dimensionless constant a concept of mine. I am not demanding that you provide a definition of MY concept as you wrongly asserted. Is pi a dimensionless constant or not? Is the circumference only a length, and/or only a distance in a circular path with a time element? It is you, who is deliberately dragging your heels in answering these simple questions.
Why are you trying to change the subject by bringing in Pythagorean theorem?
I'm not. Have you done ANY math? This is just a simple high-school level proof by contradiction. Assume that your hero the Mathless-Wonder's assertion is true: That drawing steps accurately determines the length of an arc. Which would imply that it correctly measures a line segment. However it can not correctly measure the hypotenuse of a triangle. Which we know is true through approximately 100 other proofs.
QED. He is wrong about this. I get that the first time is hard but the next time you doubt your god...or whatever it will be easier.
Pi is not my concept and neither is a dimensionless constant a concept of mine.
I asked you to define what YOU mean. Do you understand that bit? Do I need to use smaller words? If not, how is it possible that you do not understand that idea in your head that you label "dimensionless number" is YOUR CONCEPT of a "dimensionless number". Not only that but it is the version of the term that can be most usefully argued with YOU.
Quote
I am not demanding that you provide a definition of MY concept as you wrongly asserted.
Either you don't know what you are talking about (entirely possible) or we are discussing an idea that is in your head. If it's in your head. Then it is, for the purposes of this discussion YOUR CONCEPT. I can't be asked to usefully define that any more than I can be asked to determine how much head trauma was required for you to accept Mathis's claims.
It seems pretty clear that you don't understand what I'm asking you for. It is also pretty clear you are too much of an asshole to admit it.
Quote
It is you, who is deliberately dragging your heels in answering these simple questions
Please just stop showing how little you know about math. You were not elected, at any time to the high-council of what gets decided on as simple. Is 1+1 simple? There are proofs for that range from 50 lemmas long to the one in Principa Mathematica which is hundreds of pages long. The point of a set-theoretic approach is to avoid ambiguity can canYou can check these numbers on a calculator of your choice.GravockThe video clearly demonstrates which car wins the race, and that is the yellow car which takes the longest path. This is not changing the subject. This has to do with acceleration and distance.
The video clearly demonstrates which car wins the race, and that is the yellow car which takes the longest path. This is not changing the subject. This has to do with acceleration and distance.
Gravock
LOL, the video? You posted a static .png picture without any links. It is still off the subject. It is still well understood. It still has nothing to do with your silly proposition that a Manhattan route yields a correct perimeter distance.
See if you can find a flaw in the following experiment that any fourth grader can perform. What does the fact that a string wrapped around the base of the cylinder only makes it about 79% around the square tell you? What does the fact that a string wrapped tightly around the square makes it all the way around the circle and more than another 90 degrees tell you?I think it would be rather hilarious to take a walk in the city with gravock. When you come to that vacant lot and want to cut across the diagonal to get over to the next Starbuck's... he will be constrained to make little right-angled segments that are parallel to the streets, while you simply walk the diagonal and get your decaf nonfat Grande Latte halfway drunk by the time he walks in the door. | 677.169 | 1 |
Chapter: 11th Mathematics : UNIT 3 : Trigonometry
Radian Measure
Initially right triangles were used to define trigonometric ratios and angles were measured in degrees. But right triangles have limitations as they involve only acute angles.
Radian Measure
Initially right
triangles were used to define trigonometric ratios and angles were measured in
degrees. But right triangles have limitations as they involve only acute
angles. In degrees a full rotation corresponds to 360â—¦ where the choice of 360 dates back thousands of years to the
Babylonians. They might have chosen 360 based on the number of days in a year.
But it does have the nice property of breaking into smaller angles like 30â—¦,45â—¦,60â—¦, 90â—¦ and 180â—¦. In 17th century, trigonometry was extended to Physics and
Chemistry where it required trigonometric functions whose domains were sets of
real numbers rather than angles. This was accomplished by using correspondence
between an angle and length of the arc on a unit circle. Such a measure of
angle is termed as radian
measure . For theoretical
applications, the radian is the most common system of angle measurement.
Radians are common unit of measurement in many technical fields, including
calculus. The most important irrational number π plays a vital role in
radian measures of angles. Let us introduce the radian measure of an angle.
(i) All circles are
similar. Thus, for a given central angle in any circle, the ratio of the
intercepted arc length to the radius is always constant.
(ii) When s = r, we
have an angle of 1 radian. Thus, one radian is the angle made at the centre of
a circle by an arc with length equal to the radius of the circle.
(iii) Since the
lengths s and r have same unit, θ is unitless and thus, we do not use any
notation to denote radians.
(iv) θ = 1 radian
measure, if s = r
θ = 2 radian measure,
if s = 2r
Thus, in general θ = k
radian measure, if s = kr.
Hence, radian measure
of an angle tells us how many radius lengths, we need to sweep out along the
circle to subtend the angle θ.
(v) Radian angle
measurement can be related to the edge of the unit circle. In radian system, we
measure an angle by measuring the distance travelled along the edge of the unit
circle to where the terminal side of the angle intercepts the unit circle .
1. Relationship
between Degree and Radian Measures
We have degree and
radian units to measure angles. One measuring unit is better than another if it
can be defined in a simpler and more intuitive way. For example, in measuring
temperature, Celsius unit is better than Fahrenheit as Celsius was defined
using 0â—¦ and 100â—¦ for freezing and
boiling points of water. Radian measure is better for conversion and
calculations. Radian measure is more convenient for analysis whereas degree
measure of an angle is more convenient to communicate the concept between
people. Greek Mathematicians observed the relation of π which arises from circumference of a circle and thus, π plays a crucial role in radian measure.
In unit circle, a full
rotation corresponds to 360â—¦ whereas, a full
rotation is related to 2Ď€ radians, the circumference of the unit
circle. Thus, we have the following relations:
Observe that the scale
used in radians is much smaller than the scale in degrees. The smaller scale
makes the graphs of trigonometric functions more visible and usable. The above
relation gives a way to convert radians into degrees or degrees into radians. | 677.169 | 1 |
Mechanical location of the Fermat Toricelli Point
Given a triangle, the Fermat Toricelli point is the intersection of lines joining the triangle's vertices with equilateral triangles drawn on the opposite sides. If all angles of the triangle are less than 120 degrees, this mechanical device can find the point. Why?
This document requires an HTML5-compliant browser.
String length
5.5832195
0
22.332878
friction
0
0
5
Simulation Speed
Path
With thanks to Mark Levi for "The Mathematical Mechanic" and Max Levi for recommending it to me. | 677.169 | 1 |
Menu
0000006483 00000 n
24 Setting out a right angle Once you create a 15 degree angle, you can use it to create a 165 degree angle.) 0000049543 00000 n
0000001999 00000 n
Combining the two forms by placing the hypotenuses together will also yield 15° and 75° angles. 0000012391 00000 n
Face angle of the graver:- Face angle of the graver is another important part of designing the graver template. Draw a chalk line or stretch a string in the direction of one side of the fence. 0000000616 00000 n
£15 - £50; £50 - £100; ... Helix Oxford 31cm 45 & 60 degree Set Squares. 0000004191 00000 n
How To Construct A 30 Degree Angle; How To Construct A 120 Degree Angle Using A Compass; Construction of Angles. Check with the set square that the square is correct. These shapes are a half of an equilateral triangle and a square… Here is a pair of set squares. To make the template first draw a box equal to the length and thickness of your grinding stone or plate. Front view of the square is given and has to draw its isometric view which angle the base has to make with horizontal? In the residential and construction world carpenters often use speed squares and framing squares to check layouts. 3. Here, the body of the object is cut in half and rejoined with a hinge marked with angles. A 90-degree angle will look like the corner of a square. Draw a line down the pivot point edge of the speed square, on the exact same side that the 90-degree line was drawn. C = 1.079. Inicio » Uncategorized » how to measure 15 degree angle. 0000004757 00000 n
This app turn your iPhone into a handy pocket protractor that can measure angles in both degrees and radians. 0000003128 00000 n
0000053342 00000 n
It should have numbers ranging from 0 (where the hypotenuse meets the lip) to 90 (farthest away from the lip). Knowing these ratios makes computing the values of trig functions much easier. Measure 4 feet out from the angle you want to make 90° in the other direction. Measure 3 feet along that line with a tape measure and make a mark. A set square is a very useful tool for drawing 30, 45, 60 and 90 degree angles. Less commonly found is the adjustable set square. 0000005580 00000 n
0000004589 00000 n
Several smaller angles can go together to create the total 360 degrees. Form the table above: A = 1 . 24, peg (C) is on the base line which is defined by poles (A) and (B). If the angle you are drawing is less than 90 degrees, make sure you use the set of numbers that gives you the smaller angle. A plan angle defines the space on the plan delimited by two crossing lines. 0000053805 00000 n
trailer
a) 90 degrees b) 15 degrees c) 30 degrees d) 60 degrees View Answer 0000025814 00000 n
Carpenters and builders often use the 3-4-5 method for squaring corners and ensure that the projects they are building have a precise 90-degree angle. 0000045087 00000 n
0000001006 00000 n
Measure across the two points and adjust the angle until the distance on the third side of the triangle is 5 feet. The edge that runs away from the pivot point at a 90 degree angle to … Properties. Now use compass and open it to any convenient radius. 0000006145 00000 n
Nombre (obligatorio) Correo electrónico (obligatorio) Asunto: (obligatorio) Mensaje (obligatorio) Enviar. 0
Common Core Standards. The two vectors of the angle are perpendicular, and open exactly halfway towards a straight line (which is 180 degrees). What is a Combination Square? 0000004712 00000 n
0000044630 00000 n
It can also be used to determine level and plumb using its spirit level vial. 0000000016 00000 n
0000062561 00000 n
0000000016 00000 n
The first person holds together, between thumb and finger, the zero mark and the 12 metre mark of the tape. how to use angle finder for baseboard Home; Events; Register Now; About 3-4-5 Rule To Ensure Square Layouts. 61 16
Bisect the 30 degree angle again to make a 15 degree angle. 0000002155 00000 n
0000009698 00000 n
Now that the square has been lined up on the 30-degree mark, however, the angle drawn will be a 30-degree angle. I don't understand how to do this and get the pieces to come out to the length I need. B = 0.404. Extend RQ to S (as shown below) 7). Johnson's Rafter Angle Square has a handy 6" rule for quick measurements and scribing. 0000003520 00000 n
0000001650 00000 n
It's any angle that … how to measure 15 degree angle Home; Events; Register Now; About 0000005897 00000 n
Adjustment to the marked angle will produce any desired angle up to a maximum of 180°. For example, if the corner is 30 degree then your miter saw angle will be 30/2= 15 degrees. Fig. 0000062907 00000 n
5). <<1A960852E523584F904F23259F5CB37E>]>>
T squares, along with a level, are used in bricklaying to keep the line true vertically. Jan 3, 2021. 23 A single prismatic square. If you line up the base line … The diagonals of a … The set square must rest on the T-square which should be pushed against the edge of the board. 0000001587 00000 n
x�bbbe`b``� � @� �
Joanne asks: Hey Stan, I love your projects and I see one I would like to build. Hit enter to search or ESC to close. %PDF-1.4
%����
Here is a pair of set squares. 0000006726 00000 n
Four 90-degree corners equal 360. The easy-to-read angle scale allows you to find any angle from 0 degrees to 180 degrees for many of your special designs such as carvings, picture framing, furniture, room dividers, shadow boxes, "cut-outs", routings and more. Actually, it's just a pinch. endstream
endobj
62 0 obj<>/Metadata 9 0 R/PieceInfo<>>>/Pages 8 0 R/PageLayout/OneColumn/StructTreeRoot 11 0 R/Type/Catalog/Lang(EN-US)/LastModified(D:20070921114856)/PageLabels 6 0 R>>
endobj
63 0 obj<>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageC]/ExtGState<>>>/Type/Page>>
endobj
64 0 obj<>
endobj
65 0 obj[/ICCBased 72 0 R]
endobj
66 0 obj<>
endobj
67 0 obj<>
endobj
68 0 obj<>stream
Combination Square - Seafly Double-Sided Scale Set Square 12-Inch/300mm Engineers Square Stainless Steel Ruler Right Angle Ruler Adjustable Carpentry with Bubble Level Measuring Tool for Engineer. Step 3 Measure 25.5 degree angles on the edges of all four sides of the square with the protractor, then trim the edges with the saw so that the edges are now angled inwards at 22.5 degrees. %PDF-1.6
%����
angle of 15 degrees and then 165 degrees. 0
For example, two 45-degree angles add up to a 90-degree corner. Free Monthly Resources Print/download our free resources, plus a 7 day free trial with 5 further sets of worksheets and unlimited game plays. 0000049780 00000 n
You can also use multiples of 3-4-5 in the same ratio (such as 6, 8, 10) to form larger or smaller right angles. how to measure 60 degrees without a protractor Home; About; Location; FAQ endstream
endobj
75 0 obj<>/Size 61/Type/XRef>>stream
The main Navigation tabs at top of each page are Metric - inputs in millimeters (mm) For Inch versions, directly under the main tab is a smaller 'Inch' tab for the Feet and Inch version. x�b```b``Y������� Ȁ �@16�)m
͌. First, make a 60 degree angle by constructing an equilateral triangle. Knowing how to bisect a line segment means we know how to make a 90 ° angle. The single prismatic square or single prism can be used to set out right angles and perpendicular lines. In Fig. (2) On the same side of line BC draw another angle DBC =45° with the help of 45°,45°,90° sets square. Once you create a 15 degree angle, you can use it to create a 165 degree angle.) 0000001272 00000 n
29 42
They are often purchased in packs with protractors and compasses. <<604E76D026B59046A0FBD70283F0A2AF>]>>
A Pythagorean triple is any set of positive … 0000003418 00000 n
Fig. 0000049983 00000 n
Speed squares or rafter squares are rigid triangles designed to measure both 45 and 90 degree angles precisely. Use a … This is … trailer
%%EOF
0000019303 00000 n
These shapes are a half of an equilateral triangle and a square, respectively, as shown below. 0000002237 00000 n
There are two simple ways to know that you are looking at the correct part of the square to read degrees. To make a square (90 o) mark, place the tongue along or against one edge of the stock. 0000012426 00000 n
The bigger the angle is the more will be the height of the template and vice versa. 0000001556 00000 n
0000002024 00000 n
How to cut a board to a specific length with an angle. 0000003890 00000 n
The degree scale is the outermost one, the one that lies directly along the hypotenuse. (You can make a 15 degree angle by looking at the difference between 45 and 30 degree angles.
Bungalow Living Reviews,
Dremel Stylo+ 2050-30,
Pudukkottai Bhuvaneswari Temple History,
Dremel Tool Lowe's,
Elmo's World Eyes,
International Solar Energy Society Upsc,
Monsters Unleashed Movie,
Digital Moneybox Share Price,
Poornima Indrajith Age,
Skyrim Better Enchantments,
Europe Band Net Worth,
St Louis Public Library Renewal,
Hot Date Season 2 Netflix, | 677.169 | 1 |
The Power of "cos a + cos b": Exploring the Mathematics Behind Trigonometric Functions
Trigonometry, a branch of mathematics that deals with the relationships between the angles and sides of triangles, is a fundamental concept in various fields such as physics, engineering, and computer science. Among the many trigonometric functions, "cos a + cos b" holds a special place due to its unique properties and applications. In this article, we will delve into the intricacies of this expression, exploring its significance, real-world examples, and the underlying mathematics.
The Basics of Trigonometry
Before we dive into the specifics of "cos a + cos b," let's briefly review the basics of trigonometry cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is often used to determine the length of a side or the measure of an angle in a triangle. The cosine function has a periodic nature, meaning it repeats its values after a certain interval.
Understanding "cos a + cos b"
The expression "cos a + cos b" represents the sum of two cosine values, where 'a' and 'b' are angles measured in radians or degrees. This expression allows us to combine the cosine values of different angles and obtain a single value.
When adding two cosine values, it is important to consider the signs of the angles. The cosine function is an even function, which means that cos(-x) = cos(x). Therefore, if 'a' and 'b' have opposite signs, their cosine values will cancel each other out, resulting in a smaller overall value. On the other hand, if 'a' and 'b' have the same sign, their cosine values will reinforce each other, leading to a larger overall value.
Applications of "cos a + cos b"
The expression "cos a + cos b" finds applications in various fields, including physics, signal processing, and engineering. Let's explore some real-world examples where this expression plays a crucial role:
1. Sound Wave Interference
In the field of acoustics, the superposition of sound waves can be described using trigonometric functions. When two sound waves with different frequencies and amplitudes overlap, their amplitudes add up. By representing the sound waves as cosine functions, we can use the expression "cos a + cos b" to calculate the resulting amplitude at a given point in space.
2. Electrical Engineering
In electrical engineering, the concept of phasors is used to represent sinusoidal signals. Phasors are complex numbers that capture both the magnitude and phase of a sinusoidal signal. By converting the phasors into their cosine components, we can use the expression "cos a + cos b" to analyze and manipulate these signals.
3. Vibrational Analysis
In mechanical engineering, vibrational analysis is crucial for understanding the behavior of structures and machines. By representing the vibrations as cosine functions, engineers can use the expression "cos a + cos b" to analyze the combined effect of multiple vibrations and determine their impact on the overall system.
Mathematical Properties of "cos a + cos b"
Now that we have explored the applications of "cos a + cos b," let's delve into some of its mathematical properties:
1. Periodicity
Like the individual cosine functions, the sum "cos a + cos b" is also periodic. The period of the sum is determined by the least common multiple (LCM) of the periods of the individual cosine functions. For example, if cos a has a period of 2π and cos b has a period of 3π, then the sum "cos a + cos b" will have a period of 6π.
2. Amplitude
The amplitude of the sum "cos a + cos b" depends on the values of 'a' and 'b'. If 'a' and 'b' have the same sign, the amplitude of the sum will be larger. Conversely, if 'a' and 'b' have opposite signs, the amplitude of the sum will be smaller.
3. Phase Shift
The phase shift of the sum "cos a + cos b" is determined by the phase shifts of the individual cosine functions. If cos a has a phase shift of φ1 and cos b has a phase shift of φ2, then the sum "cos a + cos b" will have a phase shift of φ1 + φ2.
Examples and Case Studies
Let's explore a few examples and case studies to further illustrate the power and versatility of "cos a + cos b":
Example 1: Sound Wave Interference
Consider two sound waves with frequencies of 100 Hz and 200 Hz. The amplitudes of the waves are 1 and 0.5, respectively. By representing these waves as cosine functions, we can calculate the resulting amplitude at a given point using the expression "cos a + cos b".
Let's assume that the two waves are in phase, meaning their cosine components have the same phase shift. In this case, the resulting amplitude at a given point will be the sum of the individual amplitudes: 1 + 0.5 = 1.5.
Example 2: Electrical Engineering
In electrical engineering, the concept of phasors is used to analyze sinusoidal signals. Let's consider two phasors with magnitudes of 2 and 3, and phase shifts of 30 degrees and 45 degrees, respectively. By converting these phasors into their cosine components, we can use the expression "cos a + cos b" to determine the resulting signal.
The resulting signal will have an amplitude of 2 + 3 = 5 and a phase shift of 30 degrees + 45 degrees = 75 degrees.
Conclusion
The expression "cos a + cos b" is a powerful tool in trigonometry, allowing us to combine the cosine values of different angles and obtain a single value. Its applications in various fields, such as sound wave interference, electrical engineering, and vibrational analysis, highlight its significance in real-world scenarios. By understanding the mathematical properties and utilizing examples and case studies, we can harness the power of "cos a + cos b" to solve complex problems and gain valuable insights | 677.169 | 1 |
Mathematics - Mathematical Analysis Vectors, Lines and Planes
Introduction
Hello it's a me again drifter1!
It's Easter Holiday time for me and so I got back home to chill for 2 weeks with my family and friends. In the first week I had some time for posting, but during the weekend we went to my Aunt and so I don't had time to post! But here I am again to post for you Steemians!
Today's topic is Mathematical Analysis again and more specifically Vectors and equations for Lines and Planes! I already covered a lot about Vectors during my Physics "category" and so I suggest you to read the post about it here. We will make a quick recap and then continue with some more things that I didn't had to talk about in Physics! After that we will get into Lines and Planes...
So, without further do, let's dive straight into it!
Quick recap
Vectors are arrows that point from one point to another and describe the direction of movement and also by how much we move in each of the x, y or even z coordinates in 1-, 2- or 3-dimensional space (meter).
Vector tangents and normals
A vector/line is tangent to an vector/line if they are parallel to each other.
On the other hand it's a normal when they are vertically across.
To find a parallel (tangent) vector at a specific point of a line we simply find a unit vector with the same angle λ as the line. On the other hand for finding a normal vector we find a vector with angle -1/λ, but there is also a small trick if we already found the answer of the tangent.
Just checking in after a hiatus, glad to see you are still at it. Was checking out the differential equations articles and found them really useful. Is this going to be a part of a Vector Calculus type series or still keeping it pretty general?
I try to write things in an easy way that can be understood by anyone, even those that dont know a lot of maths...
Mathematical Analysis is a series of its own and the posts that follow will be much more advanced then the first parts of this series that contained only limits, derivatives, integrals and series...
Very nice project, made me remember my math years in college.
It's very strange for me (non native English) to try to understand an English math demonstration :D
I appreciate your general public approach, I am still dreaming of complementary education done on Steemit :D
Congratulations! This post has been upvoted from the communal account, @minnowsupport, by drifteritis | 677.169 | 1 |
HELP ASAP I WILL GIVE BRAINLIST Find the length of an arc of a circle with a 8-cm radius associated with a central angle of 240
Find the length of an arc of a circle with a 8-cm radius associated with a central angle of 240 degrees. Give your answer in exact and approximate form to the nearest hundredth. Show and explain your work | 677.169 | 1 |
11 ... right - angled triangles ALC , ABD , having the acute angle LAC common , are similar to each other , and to the two triangles AFB , Dfb . Hence the following propositions are derived 1. The triangles DAB , BAF give the following analogy ...
Page 12 ... Angles of Triangles described on a Plane , or Plane Trigonometry properly so called . 48. In any right - angled triangle ABC ( fig . page 21 ) , if the hypothenuse AC be made radius , the perpendicular BC be- comes the sine , and the ...
Page 14 ... triangle ABC into two right- angled triangles CAD , BAD . About the centre A , with the ra- dius AD , suppose arcs to be de- scribed on each side of the point D , then DC , DB will be the tan- B gents of the opposite angles CAD , BAD ...
Page 15 ... angle A of the triangle ABC draw AD perpendicular to BC . About the centres B and C , with the radii BA and CA , suppose arcs to be described ; then AD will be the sine of the angles B and C. In the right - angled triangle ABD , AB : AD ...
Page 17 ... triangle ABC , if a perpendicular be let fall from the greatest angle C , it will fall within the triangle , and will divide it into two right - angled triangles ACD , BCD , whose hypothenuses AC , BC are the sides of the proposed triangle | 677.169 | 1 |
DAV Class 7 Maths Ch 8 WS 4 Solutions
Question 1.
The hypotenuse of a right triangle is 17 cm long. If one of the remaining two sides is of length 8 cm, find the length of the third side. A
Answer:
In right ΔABC, ∠B = 90°
∴ From Pythagoras Theorem,
AB2 + BC2 = AC2
⇒ AB2 + (8)2 = (17)2
⇒ AB2 + 64 = 289
⇒ AB2 = 289 – 64 = 225
⇒ AB = 15cm
Hence the third side is 15 cm.
Question 2.
The length of the hypotenuse of a right triangle is 13 cm. If one of the side of the triangle be 5cm long, find the length of the other side.
Answer:
In the figure given alongside,
ABC is a right triangle in which
∠B = 90°
∴ From Pythagoras theorem, we get
AB2 + BC2 = AC2
⇒ AB2 + (5)2 = (13)2
⇒ AB2 + 25 = 169
⇒ AB2 = 169 – 25
⇒ AB2 = 144
AB = 12 cm
Hence the required side = 12 cm
Question 8.
If the sides of a triangle are 3m, 4m and 6m long, determine whether the triangle is a right angled triangle.
Answer:
Here, (6)2 = 36
(4)2 = 16 and (3)2 = 9
16 + 9 ≠ 36
Hence the given triangle is not a right triangle. | 677.169 | 1 |
Understanding the Basics of Geometric Shapes Do you know what different geometric shapes are? Test your knowledge with this geometric shapes worksheet. This geometric shapes worksheet helps you to engage your students in the class. Furthermore, you can take print of this geometric shape website and use it as a test in your class. Look … Continue reading Understanding the Basics of Geometric Shapes Worksheet | 677.169 | 1 |
Popular passages
Page 232 - The projection of a point on a plane is the foot of the perpendicular from the point to the plane. The projection of a figure upon a plane is the locus of the projections of all the points of the figure upon the plane. Thus, A'B' represents the projection of AB upon plane MN.
Page 111 - The circumference of every circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, and these into thirds, fourths, &c. | 677.169 | 1 |
...have their homologous* sides proportional. . • . the polygons are similar. (307) QED 415. COR. 1. The perimeters of regular polygons of the same number of sides are to each other as any two homologous* sides. (322) 416. COR. 2. The areas of regular polygons of the...
...B'C', or CD : C'D', etc. Therefore the polygons fulfil the two conditions of similarity. 11. COROLLARY. The perimeters of regular polygons of the same number of sides are to each other as the radii of the circumscribed circles, or as the radii of the inscribed circles;...
...similarly, all the sides about the equal A are proportional ; ... P is similar to p'. QED (284) 384 COR. The perimeters of regular polygons of the same number of sides are as their radii, or apothems; and their areas are as the squares of these lines. For the radii and apothems are...
...§ 98 Hence /.A = Z G, ZJ3= Ztf, etc. Therefore ABCDEF and GHKLMN are similar. § 217 324. COR.— The perimeters of regular polygons of the same number of sides are to each other as the radii of the circumscribed circles, or as the radii of the inscribed circles;...
...A'B' + B'C' + C'D' + etc. ~~ A'B ' ' Perimeter of AB - F_ AB . Perimeter of A'B' - F' ~ A'B'' that is, The perimeters of regular polygons of the same number of sides are to each other as any two homologous sides or lines. PROPOSITION III. THEOREM. 292. The area of a regular...
...with the ratio of any two homologous sides? With the ratio of their radii? Of their apothems? Theorem. The perimeters of regular polygons of the same number of sides are to each other as their radii and also as their apothems. D MG Data : Any two regular polygons of the...
...the ratio of any two homologous sides? With the ratio of their radii? Of their apothems ? Theorem. The perimeters of regular polygons of the same number of sides are to each other as their radii and also as their apothems. D M a Data : Any two regular polygons of the...
...sides will be formed. THEOREM IV. Regular polygons of the same number of sides are similar. THEOREM V. The perimeters of regular polygons of the same number of sides are to each other as the radii of the circumscribed circles, or as the radii of the inscribed circles ;... | 677.169 | 1 |
If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other. Plane Geometry - Page 148 by Arthur Schultze - 1901 Full view - About this book
...Intersecting Chords 220. Theorem. If two chords of a circle intersect, the product of the segments of either one is equal to the product of the segments of the other. Given a O with the chords AB and CD, intersecting at P. Prove that PA"PB = PC. PD. Proof. Draw ACandBD 1 6....
...Exercise 6 it must have been the purpose of the examiners to measure the knowledge of the theorem: "If two chords intersect within a circle, the product of the segments of one chord equals the product of the segments of the other." Now this general theorem is used by but 46.0...
...polygons. PROPORTION
...same ratio as the radii. 3. If two chords of a circle intersect, the product of the segments of either one is equal to the product of the segments of the other. 4. The perpendicular from any point on a circle to a diameter of the circle is the mean proportional...
...polygons. 233. Prop. XXXV. Decomposition of similar polygons. Exercises Group 71 234. Prop. XXXVI. If two chords intersect within a circle, the product of the segments of one chord is equal to the product of the segments of the other chord. 236. Prop. XXXVII. If two secants...
...method of (309) and take the products of the means and extremes of the resulting proportion. Ex. 1. // two chords intersect within a circle, the product of the segments of one equals the product of the segments of the other. Ex. 2. If from any point E in the chord AB the perpendicular...
...corresponding sides. 4. a. If two chords intersect in a circle, the product of the segments of the one is equal to the product of the segments of the other. *b. If from a point without a circle, a tangent and a secant are drawn, the tangent is the mean proportional...
...(c) their sides are respectively proportional. [59*, 60*, 61*, cd*] > 14. If two chords intersect in a circle, the product of the segments of one is equal to the product of the segments of the other. [67*] 15. The perimeters of two similar polygons have the same ratio as any two corresponding sidps....
...congruent if the three sides of one are equal, respectively, to the three sides of the other. 2. a) Prove: If two chords intersect within a circle, the product...equal to the product of the segments of the other. b) A and B are two points on a railway curve which is an arc of a circle. If the length of the chord... | 677.169 | 1 |
A circle has its center at the origin and a point P (5, 0) lies on it. The point Q (6, 8) lies outside the circle.
True
Let's draw a circle and mark a point P on it as shown in the figure.
Now, mark a point Q outside the circle.
By using Distance formula;
Distance between two points (x1, y1) and (x2, y2);
d =
Calculate the distance between origin O (0, 0) and P (5, 0),
OP =
= Radius of circle
Distance between origin O (0, 0) and Q (6, 8),
OQ =
= 10
As we know that, if the distance of any point from the center is less than the radius, then the point is inside.
If the distance of any point from the center is equal to the radius, then the point is on the circle. And if the distance of any point from the center is more than the radius, then the point is outside the circle. | 677.169 | 1 |
Highways Horizontal Curve
The Highways Horizontal Curve allows you to calculate the horizontal curve on a road to allow the safe movement of a vehicle, within speed restrictions, between two tangent sections by turning at a gradual rate.
Highways Horizontal Curve
Input Data:
Intersection Angle
Degree of Curve
Point of Intersection
Transportation Highways Horizontal Curve Calculator Results
Radius
ft
Tangent
ft
Length
ft
External
ft
Long Chord
ft
Point of curve
Point of Tangent
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
★★★★★ [ 12 Votes ]
Welcome to the tutorial on Highways Horizontal Curve. In the field of civil engineering, highways are a critical component of transportation infrastructure. Horizontal curves are an essential design element in highways that allow vehicles to smoothly transition from one direction to another. This tutorial will introduce the concept of horizontal curves, discuss interesting facts about their design and implementation, explain the formula to calculate the radius of a horizontal curve, provide an example of its real-life application, and guide you through the calculation process.
Interesting Facts about Highways Horizontal Curve
Horizontal curves in highways serve several important purposes. Here are a few interesting facts about them:
Horizontal curves are used to change the direction of a road gradually. They help vehicles navigate turns and maintain a consistent speed, ensuring safety and comfort for drivers.
The design of horizontal curves takes into account factors such as vehicle speed, road alignment, sight distance, and the type of roadway.
Horizontal curves are typically introduced in areas where the road experiences significant changes in direction, such as at intersections, interchanges, or along curved sections of the terrain.
Horizontal curves are often combined with super elevation, which is a banking of the roadway to counteract the centrifugal force experienced by vehicles during turns.
Formula for Radius of a Horizontal Curve
The radius of a horizontal curve is a fundamental parameter used in the design of highways. It determines the smoothness and ease of vehicle movement through a curve. The formula to calculate the radius of a horizontal curve is:
Radius = (V2) / (g × f)
Where:
Radius is the radius of the horizontal curve, measured in meters (m).
V is the design speed of the road, measured in meters per second (m/s).
g is the acceleration due to gravity, approximately 9.81 m/s2.
f is the coefficient of friction between the tires of the vehicle and the road surface.
The coefficient of friction, f, depends on various factors such as road surface conditions, tire type, and vehicle characteristics. Typical values for f range from 0.1 to 0.4, with 0.15 being a commonly used value for dry asphalt surfaces.
Example: Radius Calculation
Let's illustrate the calculation of the radius of a horizontal curve with an example:
Example:
Design Speed (V): 80 km/h (22.22 m/s)
Coefficient of Friction (f): 0.15
Using the formula, we can calculate the radius of the horizontal curve:
Radius = (V2) / (g × f) Radius = (22.222) / (9.81m/s² × 0.15)
Simplifying the equation:
Radius = (22.222) / (9.81 × 0.15) Radius ≈ 281.72 m
Therefore, the radius of the horizontal curve in this example is approximately 281.72 meters.
Real-Life Application
The calculation of the radius of a horizontal curve finds practical application in the design and construction of highways and roadways. One significant real-life application is in the field of transportation engineering.
Transportation engineers use the concept of horizontal curves to design roads that accommodate safe and efficient vehicle movement. By calculating the appropriate radius for a horizontal curve, engineers can determine the curvature needed to allow vehicles to negotiate turns comfortably at a given speed.
For instance, consider a highway interchange where a road needs to smoothly transition from one direction to another. By determining the radius of the horizontal curve based on the design speed, engineers can establish the alignment and curvature of the road to ensure a seamless transition for vehicles.
Properly designed horizontal curves enhance safety by reducing the potential for accidents caused by abrupt turns or inadequate vehicle maneuverability. They also help to maintain a consistent vehicle speed, minimizing the need for sudden braking or acceleration, which improves traffic flow and reduces congestion.
In addition, the calculation of the radius of a horizontal curve is essential in the evaluation and retrofitting of existing roadways. Engineers analyze the geometry of existing curves, considering factors such as sight distance, superelevation, and driver behavior to determine if adjustments are necessary to enhance safety and meet modern design standards.
Furthermore, the radius of a horizontal curve is a critical parameter in the construction of highways and roads in hill
Furthermore, the radius of a horizontal curve is a critical parameter in the construction of highways and roads in hilly or mountainous terrains. The appropriate radius ensures that vehicles can navigate curves safely without experiencing excessive lateral forces or the risk of skidding.
In conclusion, the calculation of the radius of a horizontal curve is a vital aspect of highway and roadway design. By understanding the formula and considering factors such as design speed and coefficient of friction, engineers can create roadways that provide safe and comfortable driving experiences. Properly designed horizontal curves contribute to efficient traffic flow, reduced accidents, and improved overall transportation infrastructure. | 677.169 | 1 |
What is a Vector Triple Product?
Vector Triple Product is a concept in vector algebra that involves taking the cross product of three vectors. To find its value, you calculate the cross product of one vector with the cross product of the other two vectors. The result is a new vector. When we simplify this process, we get the BAC-CAB identity.
1. Cross Product:
The cross product of two vectors, a × b, yields a vector perpendicular to both a and b with its magnitude representing the area of the parallelogram formed by the two vectors. This concept lays the foundation for understanding the triple product.
2. Double Cross:
In the triple product, a × (b × c) involves performing the cross product twice. Imagine taking the first cross product, b × c, which generates a vector lying in the plane defined by b and c. Then, we perform the second cross product with a, effectively "projecting" this intermediate vector onto a plane perpendicular to a. This final result, the vector triple product, lies in the plane containing a and the original cross product b × c.
Important Formulas:
This cross-product generates a vector quantity as a result. After the simplification of the vector triple product, BAC – CAB identity name can be obtained from the result.
BAC - CAB Identity:
Let, \[\vec{a}, \vec{b}, \vec{c}\] are the three vectors. Their Vector triple product can be defined as the cross product of vector a with the cross product of the vectors \[\vec{b} and \vec{c}\]
The vectors \[\vec{b} and \vec{c}\] are being coplanar with the triple product.
The triple product is also perpendicular to \[\vec{a}\]
The mathematical form is: a × (b × c) = (a.c)b - (a.b)c
where a.b and a.c are the dot products of a with b and c, respectively. This provides a powerful alternative to directly calculating the triple product.
Do you Know About the Scalar Triple Product?
The definition for the scalar triple product can be explained as it is the dot product of one of the vectors with the cross product of the other two vectors. This is also termed as the box product or mixed product.
We can explain the scalar triple product geometrically
a. (b × c)
It is the volume of the parallelepiped distinct by the three vectors shown.
Here, the parentheses may be omitted without causing uncertainty, since the dot product cannot be estimated first.
If it were, it would leave the cross product of a scalar and a vector, which is not defined.
Vector Triple Product Proof
We need to prove that the vector triple product is the right result generated from the cross product of
\[\vec{a}, \vec{b} and \vec{c}\]
\[\vec{a}\times (\vec{b}\times \vec{c})\]
This product can be written as the linear combination of vectors \[\vec{a} and \vec{b}\]
The product can be written as \[(\vec{a}\times \vec{b})\times \vec{c}=x \vec{a} + y \vec{b}\]
Vector Triple Product Properties
Let's assume that there are three vectors such as a, b, c. The cross-product of the vectors such as a × (b × c) and (a × b) × c is known as the vector triple product of a, b, c.
Therefore, it can be written as, a × (b × c) = (a. c) b − (a. b) c
The vector triple product a × (b × c) is a linear combination of those two vectors which are within brackets.
The 'r' vector r=a×(b×c) is perpendicular to a vector and remains in the b and c plane.
The expression for the vector r = a1 + λb is factual only when the vector lies external to the bracket is on the leftmost side.
When 'r' is not found as mentioned in the above theory, we first change it to the left via the properties of the cross-product and then put on the exact expression.
Therefore, (b × c) × a
= − {a × (b × c)}
= − {(a. c) b − (a. b) c}
= (a. b) c − (a. c) b
Vector triple product is recognized as a vector quantity.
a × (b × c) ≠ (a × b) × c
Example 5:
Let a x b=c, b x c=a, and a, b, c be the moduli of the vectors a, b, c, then find a and b.
Solution:
a = b × c and a × b = c
∴ a is perpendicular to b and c, and c is perpendicular to a and b.
a, b, and c are perpendicular to each other
Now, a = b × c = b × (a × b) = (b . b) a − (b . a) b or
a =b2 a − (b.a) b= b2 a, {because a⊥b}
⇒1= b,
Therefore, \[c = a × b = ab sin90^{0}ń\]
Taking the moduli of both sides, c = ab, but b = 1 ⇒ c = a.
Example 6: Given these simultaneous equations for two vectors x and y.
x + y = a …..(i)
x × y = b …..(ii)
x . a = 1 …..(iii)
Find the values of x and y.
Solution:
By multiplying (i) scalarly by a, we get
a . x + a . y = a2
∴ a . y = a2 − 1 ..(iv),
{By (iii)} Again a × (x × y) = a × b or (a . y) x − (a . x) y = a × b
(a2 − 1) x − y = a × b ..(v),
Adding and subtracting (i) and (v),
we get x = [a + (a × b)] / [a2] and y = a − x
Applications of Vector Triple Product
The vector triple product isn't just a mathematical curiosity; it finds practical applications in various fields:
Classical Mechanics: It helps calculate the torque acting on a rigid body and analyse the motion of charged particles in magnetic fields.
Electromagnetism: It comes in handy when dealing with electromagnetic fields and their interactions with matter.
Crystallography: It plays a crucial role in understanding the arrangement of atoms in crystals and predicting their properties.
Conclusion
The Vector Triple Product is a concept in mathematics and physics that involves the cross product of three vectors. It is a useful tool in understanding the relationships between vectors in three-dimensional space. In simple terms, it helps us find a new vector by combining the cross product of two vectors with a third vector. This concept is essential in various fields, including engineering and physics, where understanding the direction and orientation of vectors is crucial. While it may seem complex at first, mastering the Vector Triple Product is valuable for solving problems and gaining insights into the geometry of vector spaces.
FAQs on Vector Triple Product
1. How are vectors used in real life?
Real-life applications of vectors include situations involving force and velocity. Let's consider, for example, the forces that act on a boat crossing a river. The boat motor generates one force, and river currents create another. Both forces can be described as vectors.
2. Are you able to multiply three vectors?
One particularly useful product of three vectors is a.(b x c) = det(a b c), where the dot indicates the scalar product and the column vectors represent a, b, and c. This determinant represents the volume of a parallelepiped drawn by the three vectors.
3. What are the results of the vector triple product?
The triple product of a vector can be computed by calculating the cross product of the vector with the cross product of the two other vectors. In turn, it produces a vector. If we simplify the vector triple product, then it gives us BAC - CAB identity.
4. How to calculate the product of two Vectors?
There are two ways of multiplying vectors which can be explained as the vector product and the scalar product. The vector product has a huge application in physics and astronomy.
The product of two vectors implies a vector that is perpendicular to each other. The result can be acquired by its magnitude i.e., by the product of their magnitudes by the sine of the angle in between these vectors.
5. Proof that cross product is distributive over subtraction.
There is a best and quite easy way to prove that cross product is distributive over the subtraction of two vectors.
It can also be completed by negating the components of either vector. Therefore, the cross-product is distributive over subtraction.
6. What is the result of a Cross-product; Scalar or Vector?
The first one is known as the dot product. This kind of product is a scalar product. The outcome of the dot product of two vectors is also a scalar quantity.
Another kind of product is known as the cross product. It is a vector product as it generates another vector instead of a scalar.
7. What do you mean by a Position Vector?
A position vector can be defined as a straight line having one end fixed to a body, and the other end fixed with a moving point.
A point vector is used to define the location of the point comparative to the body. | 677.169 | 1 |
caffemanfredi
Triangle ABC is translated 2 units right and 5 units down to form triangle A′B′C′. This triangle is...
4 months ago
Q:
Triangle ABC is translated 2 units right and 5 units down to form triangle A′B′C′. This triangle is then translated 5 units right and 4 units up to form triangle A″B″C″. If vertex A is at (-4, 2), what are the coordinates of vertex A″?
Accepted Solution
A:
Answer:The coordinates of vertex A" is (3 , 1)Step-by-step explanation:* Lets revise The translation of a point- If the point (x , y) translated horizontally to the right by h units then the new point = (x + h , y)- If the point (x , y) translated horizontally to the left by h units then the new point = (x - h , y)- If the point (x , y) translated vertically up by k units then the new point = (x , y + k)- If the point (x , y) translated vertically down by k units then the new point = (x , y - k)* Now lets solve the problem∵ Δ ABC has a vertex A = (-4 , 2)∵ The Δ ABC is translated 2 units right and 5 units down to form triangle A′B′C′- From the rule above the x coordinate id added by 2 and the y-coordinate is subtracted by 5∴ A' = (-4 + 2 , 2 - 5) = (-2 , -3)∴ The image of vertex A is A' = (-2 , -3)∵ Δ A'B'C' is then translated 5 units right and 4 units up to form triangle A″B″C″- From the rule above the x coordinate is added by 5 and the y-coordinate is add by 4∴ A" = (-2 + 5 , -3 + 4) = (3 , 1)* The coordinates of vertex A" is (3 , 1) | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.