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Euler angles are three angles used to describe the orientation of a rigid body, they are typically denoted $\alpha, \beta, \gamma$, these angles represent a sequence of three elemental rotations about the axes of some coordinate system
Intrinsic and extrinsic rotations
Intrinsic rotations
A set of intrinsic rotations represent rotations relative to the object space which changes after each rotation
If the axes of some coordinate system are $X,Y,Z$ (note that initially the axes are aligned with the axes of a fixed coordinate system $x,y,z$), one of the most conventional set of intrinsic rotations is $z-x'-z''$, it's computed'$-axis, the resulting set of axes is $x'', y'', z''$ (note that $x'' = x'$)
perform a rotation of $\gamma$ around the $z''$-axis, the resulting set of axes is $x''', y''', z'''$ (note that $z''' = z''$ and that the object space $z$-axis is used twice in the overall rotation)
intrinsic rotation \(z-x'-z''\), note that the \(+z\)-axis points upward, the \(+x\)-axis points left and the \(+y\)-axis point right (all shown in blue), the rotated system \(X,Y,Z\) is shown in red
A rotation matrix (used to pre-multiply column vectors) can be used to represent a sequence of intrinsic rotations, for example the extrinsic rotations $x-y'-z''Extrinsic rotations
A set of extrinsic rotations represent rotations relative to a fixed coordinate system (typically the world coordinate system), for example the set of extrinsic rotations $z-x-z$ works$-axis, the resulting set of axes is $x'', y'', z''$
perform a rotation of $\gamma$ around the $z$-axis, the resulting set of axes is $x''', y''', z'''$
A rotation matrix (used to pre-multiply column vectors) can be used to represent a sequence of intrinsic rotations, for example the extrinsic rotations $x-y-zConversion between intrinsic rotations and extrinsic rotations
Any intrinsic rotation is equivalent to an extrinsic rotation by the same angles but with inverted order of rotations
For example the intrinsic rotations $x-y'-z''$ by the angles $\alpha,\beta,\gamma$ are equivalent to the extrinsic rotations $z-y-x$ by the angles $\gamma,\beta,\alpha$, both represented by
Proper Euler angles
A sequence of three elemental rotations are called proper Euler angles when the first and third rotation axes are the same
Proper Euler angles representing rotations about \(z-x'-z''\) by the angles \(\alpha, \beta, \gamma\), the rotated system \(X,Y,Z\) is shown in red
There are six possibilities of choosing the rotation axes for proper Euler angles which are intrinsic rotations in a similar way there are other six other possibilities of choosing the rotation axes which are extrinsic rotations
intrinsic rotations
extrinsic rotations
$x-y'-x''$
$x-y-x$
$x-z'-x''$
$x-z-x$
$y-x'-y''$
$y-x-y$
$y-z'-y''$
$y-z-y$
$z-x'-z''$
$z-x-z$
$z-y'-z''$
$z-y-z$
Tait-Bryan angles
A sequence of three elemental rotations are called Tail-Bryan angles when the angles represent rotations about three distinct axes
Just like proper Euler angles there are 6 possible intrinsic rotations and 6 possible extrinsic rotations
intrinsic rotations
extrinsic rotations
$x-y'-z''$
$z-y-x$
$x-z'-y''$
$y-z-x$
$y-x'-z''$
$z-x-y$
$y-z'-x''$
$x-z-y$
$z-x'-y''$
$y-x-z$
$z-y'-x''$
$x-y-z$
The set of intrinsic rotations $z-y'-x''$ is known as yaw, pitch and roll, these angles are also known as nautical angles because they can describe the orientation of a ship or aircraft
Tait–Bryan angles representing the sequence \(z-y'-x''\)
The rotation matrix for the sequence $z-y'-x''$ (or $x-y-z$) which is known as yaw, pitch and roll is given by
Extrinsic rotations expressed in upright space
An important thing to note is that the standard rotation matrices work in upright space, if the object space axes are not aligned with the upright space axes (different direction) then the sequence of extrinsic rotations must be done on the axes expressed in upright space
For example given that world space is
Chosen world space \(+x\) (right), \(+y\) (up) and \(+z\) (backward), note that the choice is just personal preference
If there's an object whose object space axes $+x$ (backward), $+y$ (right) and $+z$ (up) then a sequence of intrinsic rotations $z-y'-x''$ by the angles $\alpha, \beta, \gamma$ (equivalent to the extrinsic rotation $x-y-z$ by the angles $\gamma, \beta, \alpha$ which is also known as yaw, pitch and roll) is equivalent to the multiplication of the following rotation matrices
The problem can be simplified when frame is somewhat aligned with the upright space (the order might be different and the axis directions might be reversed but it's still aligned), the following diagram shows some of these simplifications | 677.169 | 1 |
Unit 5 trigonometric functions homework 4 answer key
Real World Applications of SOHCATOA. This result should not be surprising because, as we see from Figure 59, the side opposite the angle of π 3 is also the side adjacent to π 6, so sin( π 3) and cos( π 6) are exactly the same ratio of the same two sides, √3s and 2s. Probably the most familiar unit of angle measurement is the degree. Lesson 6 - Arithmetic Sequences Practice Test Answer Key. ADJ the side adjacent to the angle θ. Inverse Trig Functions Worksheet (pdf) and Answer Key Students will practice determing angles of right triangles by using inverse trig functions-- arcsine, arc cos, arc tan. The ambiguity of Sines is explored. One of the standout featu.
Find an angle with a given terminal point. Find step-by-step solutions and answers to Pre-Calculus 12 Student Workbook - 9780070738911, as well as thousands of textbooks so you can move forward with confidence Chapter 4:Trigonometry and the Unit Circle1: Angles and Angle Measure2: The Unit Circle Chapter 5:Trigonometric Functions and Graphs1. 7_simplifying_ration_functions_w_answer_key. Like any other printer, it requires a driver to function properly When it comes to purchasing a new pillow, one of the key factors to consider is the warranty that comes with it. • sin = • sin R 14 50 tan R • tan = Directions: Solve for x.
Unit 7: Right Triangle Trigonometry. When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see Figure 2).
, the secant of angle t is equal to 1 cost = 1 x, x ≠ 0. Simplify csc2Ð cos2Ðcsc2Ð = 1 There are two different ways you can leave this answer! In the notes, leave itin terms ofsin2Ð. | 677.169 | 1 |
Hint: Here the eccentricity of an ellipse is a measure of how nearly circular is the ellipse. Eccentricity is found by the formula eccentricity = c/a where 'c' is the distance from the centre to the focus of the ellipse and 'a' is the distance from the centre to the vertex for the standard form of the ellipse.
Note: The eccentricity of the ellipse is always greater than zero but less than one i.e. \[0 < e < 1\]. The standard form of the ellipse is \[\dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1\] with centre \[\left( {h,k} \right)\]. In this problem we have centre \[\left( {0,0} \right)\] so we have used the ellipse form \[\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1\]. | 677.169 | 1 |
Trigonometry and Trigonometric Functions
This set of MCQs helps you brush up on important math topics and prepare you to dive into skill practice.
Start Quiz
Find side AB?
5
8.66
8.76
6,87
Find the length of side w.
22
18
26.8
43.6
Find the missing angle θ.
26.8
27.3
25
22.87
Solve for the missing distance.
16.9
16.79
18.9
27.5
A ladder of length 4m makes an angle of 30∘ with the floor while leaning against one wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60∘ with the floor. Find the distance between the two walls of the room. | 677.169 | 1 |
Calculating the Height of a Triangle: A Comprehensive Guide
In the realm of geometry, triangles stand out as one of the most fundamental shapes, possessing unique properties that have intrigued mathematicians and scientists for centuries. Among these properties, determining the height of a triangle plays a crucial role in various applications, ranging from basic geometry calculations to complex engineering designs.
This comprehensive guide delves into the intricacies of calculating the height of a triangle, providing a clear and step-by-step approach to understanding the concept and its practical applications. By exploring different methods and formulas, we aim to empower readers with the knowledge and skills necessary to tackle height calculations with confidence.
Before embarking on our exploration of height calculation methods, it is essential to establish a solid understanding of the concept of triangle height. In geometry, the height of a triangle, often denoted by 'h,' represents the perpendicular distance from the vertex of the triangle to the opposite side, also known as the base of the triangle.
Calculating Height of a Triangle
Determining the height of a triangle involves various methods and formulas. Here are eight important points to consider:
Identify Triangle Type
Use Base and Area
Apply Trigonometry
Involve Similar Triangles
Leverage Heron's Formula
Explore Pythagorean Theorem
Consider Median and Altitude
Understand Centroid and Orthocenter
By understanding these key points, you'll gain a comprehensive grasp of calculating the height of a triangle, enabling you to solve geometry problems with precision and confidence.
Identify Triangle Type
The initial step in calculating the height of a triangle is to identify its type. There are three primary types of triangles based on their angles and side lengths:
1. Equilateral Triangle:
All three sides are equal in length.
All three angles measure 60 degrees.
The height of an equilateral triangle bisects the base and is also the median and altitude.
2. Isosceles Triangle:
Two sides are equal in length.
The angles opposite the equal sides are also equal.
The height of an isosceles triangle bisects the base and is also the median and altitude.
3. Scalene Triangle:
All three sides are different in length.
All three angles are different in measure.
The height of a scalene triangle is not necessarily the median or altitude.
Once you have identified the type of triangle, you can proceed with the appropriate method to calculate its height.
Understanding the triangle type is crucial because different formulas and methods apply to each type. By correctly identifying the triangle type, you lay the foundation for accurate height calculations.
Use Base and Area
In certain scenarios, you can determine the height of a triangle using its base and area. This method is particularly useful when the triangle's height is not directly given or cannot be easily calculated using other methods.
Formula:
Height (h) = 2 * Area / Base
Steps:
Calculate the area (A) of the triangle using the appropriate formula based on the given information (e.g., base and height, side lengths and semi-perimeter, etc.).
Identify or measure the length of the base (b) of the triangle.
Substitute the values of A and b into the formula: Height (h) = 2 * Area / Base.
Simplify the expression to find the height (h) of the triangle.
Example:
Given a triangle with a base of 10 units and an area of 20 square units, calculate its height.
Using the formula: Height (h) = 2 * Area / Base
h = 2 * 20 / 10
h = 4 units
Therefore, the height of the triangle is 4 units.
Applicability:
This method is particularly useful when dealing with right triangles or triangles where the height cannot be directly obtained from trigonometric ratios or other geometric properties.
Remember, the formula Height (h) = 2 * Area / Base is specifically applicable to triangles. For other quadrilaterals or polygons, different formulas and methods are required to calculate their heights or altitudes.
Apply Trigonometry
Trigonometry offers a powerful toolset for calculating the height of a triangle when certain angles and side lengths are known. This method is particularly useful in right triangles, where trigonometric ratios can be directly applied.
Right Triangle:
In a right triangle, the height (h) is the side opposite the right angle. Using trigonometric ratios, you can calculate the height based on the known angle and side lengths.
Sine Ratio:
The sine ratio (sin) is defined as the ratio of the height (opposite side) to the hypotenuse (longest side) of a right triangle.
Formula: sin(angle) = Height / Hypotenuse
Cosine Ratio:
The cosine ratio (cos) is defined as the ratio of the base (adjacent side) to the hypotenuse of a right triangle.
Formula: cos(angle) = Base / Hypotenuse
Tangent Ratio:
The tangent ratio (tan) is defined as the ratio of the height (opposite side) to the base (adjacent side) of a right triangle.
Formula: tan(angle) = Height / Base
To calculate the height of a right triangle using trigonometry, follow these steps:
Identify the right angle and label the sides as hypotenuse, base, and height.
Measure or determine the length of one side and the measure of one acute angle.
Use the appropriate trigonometric ratio (sine, cosine, or tangent) based on the known information.
Substitute the values into the trigonometric equation and solve for the height (h).
Remember that trigonometry can also be applied to non-right triangles using the Law of Sines and the Law of Cosines, but these methods are more advanced and require a deeper understanding of trigonometry.
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Leverage Heron's Formula
Heron's Formula provides a versatile method for calculating the area of a triangle using its side lengths. Interestingly, this formula can be巧妙地crafted to derive the height of a triangle as well.
Heron's Formula:
Area (A) = √[s(s – a)(s – b)(s – c)]
where 's' is the semi-perimeter of the triangle and 'a', 'b', and 'c' are the lengths of its sides.
Derivation for Height:
Rearranging Heron's Formula, we get:
h = 2A / b
where 'h' is the height of the triangle, 'A' is the area, and 'b' is the length of the base.
Steps to Calculate Height:
Calculate the semi-perimeter (s) of the triangle: s = (a + b + c) / 2
Compute the area (A) of the triangle using Heron's Formula.
Identify the base (b) of the triangle, which is the side perpendicular to the height.
Substitute the values of A and b into the formula: h = 2A / b.
Simplify the expression to find the height (h) of the triangle.
Example:
Given a triangle with sides of length 6 units, 8 units, and 10 units, calculate its height if the base is the side with length 8 units.
Step 1: Semi-perimeter (s) = (6 + 8 + 10) / 2 = 12
Step 2: Area (A) = √[12(12 – 6)(12 – 8)(12 – 10)] = 24√2 square units
Step 3: Base (b) = 8 units
Step 4: Height (h) = 2A / b = 2(24√2) / 8 = 6√2 units
Therefore, the height of the triangle is 6√2 units.
Heron's Formula offers a convenient way to calculate the height of a triangle, especially when the side lengths are known and the height cannot be directly obtained using other methods. It's worth noting that Heron's Formula can also be applied to calculate the area of a triangle, making it a versatile tool for various geometric problems.
Explore Pythagorean Theorem
The Pythagorean Theorem is a cornerstone of geometry, providing a powerful tool for calculating the height of a right triangle. This theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Pythagorean Theorem:
a² + b² = c²
where 'a' and 'b' are the lengths of the two shorter sides (legs) and 'c' is the length of the hypotenuse.
Derivation for Height:
In a right triangle, the height (h) is the altitude from the right angle to the hypotenuse. Using the Pythagorean Theorem, we can derive a formula for the height:
h² = c² – b²
where 'h' is the height, 'c' is the length of the hypotenuse, and 'b' is the length of the base (the side adjacent to the height).
Steps to Calculate Height:
Identify the right triangle and label the sides as hypotenuse, base, and height.
Measure or determine the lengths of the hypotenuse and the base.
Substitute the values of 'c' and 'b' into the formula: h² = c² – b².
Simplify the expression to solve for 'h²'.
Take the square root of 'h²' to find the height (h) of the triangle.
Example:
Given a right triangle with a hypotenuse of 10 units and a base of 6 units, calculate its height.
Step 1: h² = 10² – 6² = 64
Step 2: h = √64 = 8 units
Therefore, the height of the triangle is 8 units.
The Pythagorean Theorem provides a straightforward method for calculating the height of a right triangle, especially when the lengths of the hypotenuse and base are known. It's worth noting that this theorem is only applicable to right triangles, and for other types of triangles, different methods may need to be employed.
Consider Median and Altitude
In the realm of triangle geometry, the median and altitude offer valuable insights into the triangle's structure and properties. While they are closely related, they serve distinct purposes in calculating the height of a triangle.
Median:
A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. It divides the triangle into two equal areas.
Altitude:
An altitude is a line segment that extends from a vertex of a triangle perpendicular to the opposite side. It is also known as the height of the triangle.
Relationship between Median and Altitude:
In an isosceles triangle, the median and altitude coincide, meaning they are the same line segment. This is because the base angles of an isosceles triangle are equal, and the altitude bisects the base, making it also a median.
Calculating Height using Median:
In an isosceles triangle, the median (which is also the altitude) can be used to calculate the height. The formula is:
Height (h) = √(a² – (b/2)²)
where 'a' is the length of one of the equal sides and 'b' is the length of the base.
It's worth noting that the median-based formula is only applicable to isosceles triangles. For scalene triangles, where all sides are of different lengths, the median and altitude do not coincide, and different methods are required to calculate the height.
Understand Centroid and Orthocenter
In the study of triangle geometry, the centroid and orthocenter are two significant points that provide insights into the triangle's properties and characteristics.
Centroid:
The centroid of a triangle is the intersection point of its three medians. It is also known as the geometric center or barycenter of the triangle.
Orthocenter:
The orthocenter of a triangle is the intersection point of its three altitudes. It is the point where all three altitudes meet.
Relationship between Centroid and Orthocenter:
In an equilateral triangle, the centroid and orthocenter coincide, meaning they are the same point. This is because the altitudes and medians of an equilateral triangle are all congruent and intersect at the same point.
Calculating Height using Centroid:
In an equilateral triangle, the height can be calculated using the distance between the centroid and any vertex.
Height (h) = √(3/4) * side length
It's important to note that the centroid and orthocenter do not generally coincide in scalene and isosceles triangles. The orthocenter may lie inside, outside, or on the triangle, depending on the specific triangle.
FAQ
Introduction:
To complement your understanding of calculating triangle heights, here's a comprehensive FAQ section addressing common questions related to using a calculator for these calculations.
Question 1: Can I use a calculator to find the height of a triangle?
Answer: Yes, you can use a calculator to find the height of a triangle. In fact, calculators are particularly useful when dealing with complex calculations or when the values involved are large or have decimal places.
Question 2: Which formula should I use on my calculator to find the height of a triangle?
Answer: The formula you use will depend on the information you have about the triangle. Common formulas include:
Question 3: What if I don't know all the side lengths or angles of the triangle?
Answer: If you don't have all the necessary information, you may need to use trigonometric ratios (sine, cosine, tangent) to find the missing values. These ratios can be calculated using the known sides and angles.
Question 4: How do I handle square roots or complex calculations on my calculator?
Answer: Most scientific calculators have a square root function (√) and other mathematical functions built in. Simply follow the instructions provided in your calculator's manual to perform these calculations.
Question 5: Can I use a calculator to find the height of equilateral, isosceles, or scalene triangles?
Answer: Yes, you can use a calculator for any type of triangle. However, you may need to use different formulas or methods depending on the specific triangle type.
Question 6: Are there any online calculators available for finding the height of a triangle?
Answer: Yes, there are various online calculators that can help you find the height of a triangle. Simply enter the known values into the calculator, and it will provide you with the result.
Closing:
By understanding these frequently asked questions, you can confidently use your calculator to find the height of a triangle, regardless of the given information or triangle type. Remember to always check your calculator's instructions for specific functions or operations.
Now that you have a better understanding of using a calculator for triangle height calculations, let's explore some additional tips to make the process even smoother.
Tips
Introduction:
To enhance your skills in calculating triangle heights using a calculator, here are four practical tips to make the process more efficient and accurate:
Tip 1: Choose the Right Calculator:
Not all calculators are created equal. For complex calculations involving trigonometric functions or square roots, it's best to use a scientific calculator. These calculators have built-in functions that can handle these operations easily.
Tip 2: Understand the Formula:
Before using your calculator, make sure you understand the formula you'll be using to calculate the height. Having a clear grasp of the formula will help you enter the correct values and interpret the result accurately.
Tip 3: Organize Your Work:
To avoid errors, organize your work neatly. Label the given values and the intermediate steps of your calculation. This will help you keep track of your progress and identify any potential mistakes.
Tip 4: Double-Check Your Work:
Once you've obtained a result, it's always a good practice to double-check your work. Recalculate the height using a different method or an online calculator to verify your answer. This extra step can save you from errors and ensure accurate results.
Closing:
By following these simple tips, you can streamline your triangle height calculations using a calculator. Remember, practice makes perfect, so the more you work with different triangles and formulas, the more proficient you'll become.
Equipped with these tips and the knowledge gained throughout this guide, you're well-prepared to tackle any triangle height calculation that comes your way. Whether you're solving geometry problems or working on engineering projects, these techniques will serve you well.
Conclusion
Summary of Main Points:
Throughout this comprehensive guide, we've explored various methods and techniques for calculating the height of a triangle. We began by emphasizing the importance of identifying the triangle type, as different types require different approaches.
We then delved into specific methods, including using the base and area, applying trigonometry, involving similar triangles, leveraging Heron's Formula, exploring the Pythagorean Theorem, and considering the median and altitude. Each method was explained in detail with clear steps and examples.
To enhance your understanding, we also provided a tailored FAQ section addressing common questions related to using a calculator for triangle height calculations. Finally, we offered practical tips to make the calculation process more efficient and accurate.
Closing Message:
With the knowledge and skills gained from this guide, you're now equipped to confidently tackle triangle height calculations in various contexts. Whether you're a student solving geometry problems, an engineer designing structures, or a professional working with triangles, this guide has provided you with a solid foundation.
Remember, practice is key to mastering these techniques. The more you work with different triangles and formulas, the more comfortable and proficient you'll become in calculating triangle heights. So, embrace the challenge, explore different problems, and enjoy the satisfaction of finding accurate solutions. | 677.169 | 1 |
Answer
$125^{\circ}$
Work Step by Step
The big hand is at $\frac{10}{60}360^{\circ}=60^{\circ}$, compared to twelve o'clock, the small hand is at $\frac{6}{12}360^{\circ}+\frac{10}{60}\frac{1}{12}360^{\circ}=180^{\circ}+5^{\circ}=185^{\circ}$. Therefore, the angle is $185^{\circ}-60^{\circ}=125^{\circ}$ | 677.169 | 1 |
How do you prove a cyclic quadrilateral is a circle?
If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form the vertices of a cyclic quadrilateral.
How do you prove the circle theorem?
Draw two radii as shown. Since an angle subtended at the circumference by an arc is half that subtended at the centre, the angles round the centre are 2a and 2b. Angles round a point add up to 360° so 2a + 2b = 360°. Therefore a + b = 180°, so the theorem is proven.
What is cyclic quadrilateral in circle?
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
What is circle theorem?
Circle theorem may refer to: Any of many theorems related to the circle; often taught as a group in GCSE mathematics. These include: Inscribed angle theorem. Thales' theorem, if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ∠ABC is a right angle.
What are theorems and proofs?
proofA proof is a series of true statements leading to the acceptance of truth of a more complex statement. is the hypotenuse of the triangle. theoremA theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.
How do you prove a quadrilateral theorem is inscribed?
Proof: In the quadrilateral ABCD can be inscribed in a circle, then we have seen above using the inscribed angle theorem that the sum of either pair of opposite angles = (1/2(a1 + a2 + a3 + a4) = (1/2)360 = 180. Conversely, if the quadrilateral cannot be inscribed, this means that D is not on the circumcircle of ABC.
How do you prove a cyclic quadrilateral is a rectangle?
Each angle of a rectangle is a right angle. For a cyclic quadrilateral, sum of opposite angles is 180°. => 90° + 90° = 180° ( sum of opposite angles of a rectangle ). Hence, rectangle is a cyclic quadrilateral.
What are the circle theorem rules?
Circle Theorems
Theorem 1: The angle in a semicircle is 90°
Theorem 2: The angle at the centre is double the angle at the circumference.
Theorem 3: Angles from the same chord in the same segment are equal.
Theorem 4: Opposite angles in a cyclic quadrilateral sum to 180 °
Theorem 5: Alternate segment theorem.
How do you prove the diameter of a circle?
Given : In circle with center O,CD is the diameter and AB is the chord which is bisected by diameter at E,OA and OB are joined. | 677.169 | 1 |
Mastering The Properties And Uses Of Equilateral Triangles In Mathematics And Real-Life Applications.
equilateral triangle
a triangle with 3 congruent sides
An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are also of equal measure, which is 60 degrees each. Some important properties of an equilateral triangle are:
1. Equal sides: All three sides of an equilateral triangle are congruent, which means they are of equal length.
2. Equal angles: All three angles of an equilateral triangle are congruent, which means they are of equal measure.
3. Perimeter: The perimeter of an equilateral triangle is the sum of the lengths of its three sides. If a side of the triangle is denoted by a, then the perimeter (P) is given by P = 3a.
4. Area: The area of an equilateral triangle can be found using the formula A = (sqrt(3)/4) x a^2, where A is the area and a is the length of a side.
5. Height: The height of an equilateral triangle is the perpendicular distance from any vertex to the opposite side. It can be found using the formula h = (sqrt(3)/2) x a, where h is the height and a is the length of a side.
6. Circumcenter: The circumcenter of an equilateral triangle is the point where the perpendicular bisectors of the sides meet. It is equidistant from all three vertices of the triangle.
7. Incenter: The incenter of an equilateral triangle is the point where the angle bisectors of the angles meet. It is equidistant from all three sides of the triangle.
Equilateral triangles have many interesting properties and are commonly used in both mathematics and real-world | 677.169 | 1 |
triangle midsegment theorem examples
Triangle Midsegment Theorem Worksheet Pdf – Triangles are among the most fundamental forms in geometry. Understanding the concept of triangles is essential for understanding more advanced geometric concepts. In this blog, we will cover the various kinds of triangles with triangle angles. We will also discuss how to calculate the perimeter and area of a triangle, and provide illustrations of all. Types of Triangles There are three types from triangles: Equal, isosceles, and scalene. … Read more
The Triangle Midsegment Theorem Worksheet – Triangles are among the fundamental shapes in geometry. Understanding triangles is vital to developing more advanced geometric ideas. In this blog post this post, we'll go over the various kinds of triangles and triangle angles, as well as how to calculate the area and perimeter of a triangle, and provide illustrations of all. Types of Triangles There are three types that of triangles are equilateral, isosceles, and scalene. Equilateral triangles consist … Read more | 677.169 | 1 |
Angle worksheets are an integral part of any Each worksheet contains six problems.
These worksheets are intended to help students understand angles better. They are great for students in grades five and higher. Some of these worksheets also contain special instructions that help students understand the concept. They can also practice finding angles on graphs not aligned. | 677.169 | 1 |
In geometric terms, the cosine of an angle returns the ratio of a right triangle's adjacent side over its hypotenuse. For example, the cosine of PI ()/6 radians (30°) returns the ratio 0.866. = COS(PI() / 6) // Returns 0.886. 2007-06-03
Cos(30) = 0.866025403784 This is the same answer you will get if you have a scientific calculator set to DEG mode and then enter 30 followed by the Cos button. To find cos of another number, please enter the number below and press "Calculate Cos".
Cos is the cosine function, which is one of the basic functions encountered in trigonometry. It is defined for real numbers by letting be a radian angle measured counterclockwise from the axis along the circumference of the unit circle. Cos [x] then gives the horizontal coordinate of the arc endpoint. The equivalent schoolbook definition of the cosine of an angle in a right triangle is the
Cos 60 + Sin 60 . Cos 30 = If playback doesn't begin shortly, try restarting your device. Videos you watch may be added to the TV's watch history and influence TV
Use this simple cos calculator to calculate the cos value for 45° in radians / degrees. The Trignometric Table of sin, cos, tan, cosec, sec, cot is useful to learn the common angles of trigonometrical ratios from 0° to 360°. Select degrees or radians in the drop down box and calculate the exact cos 45° value easily. | 677.169 | 1 |
To find the longest possible perimeter of the triangle, you need to maximize the length of the third side.
Given that two angles of the triangle are π/3 and π/4, you can use the fact that the sum of the angles in a triangle is π radians (180 degrees) to find the measure of the third angle:
Third angle = π - (π/3) - (π/4)
With this third angle, you can now determine the longest possible perimeter by maximizing the length of the third side. Since one side of the triangle has a length of 1, the longest possible perimeter occurs when the third side is as long as possible.
By using trigonometric relationships (such as the Law of Sines), you can determine the length of the third side. However, for simplicity, the triangle with angles of π/3, π/4, and the third angle will be an isosceles triangle with equal sides opposite the equal angles.
Now, you have the lengths of two sides of the triangle. Since the triangle is isosceles, the third side will also have the same length.
Calculate the length of the third side using the above calculations.
Once you have the lengths of all three sides, you can find the perimeter of the triangle by adding them together.
This perimeter will be the longest possible perimeter of the triangle | 677.169 | 1 |
CBSE 9th Maths
Areas Of Parallelograms And Triangles
Note: As per the revised CBSE curriculum, this chapter has been removed from the syllabus for the 2020-21 academic session.
The area represents the amount of planar surface being covered by a closed geometric figure.
Area of a parallelogram
The area of a parallelogram is the product of any of its sides and the corresponding altitude.
Area of a parallelogram = b×h
Where 'b′ is the base and 'h′ is the corresponding altitude(Height).
Area of a triangle
Area of a triangle = (1/2)×b×h
Where "b" is the base and "h" is the corresponding altitude.
Theorems
Parallelograms on the Common Base and Between the Same Parallels
Theorem: Parallelograms that lie on the common base and between the same parallels are said to have equal in area.
Two parallelograms are said to be on the common/same base and between the same parallels if
a) They have a common side.
b) The sides parallel to the common side lie on the same straight line.
Triangles on the Common Base and Between the Same Parallels
Theorem: Triangles that lie on the same or the common base and also between the same parallels are said to have an equal area.
Here, ar(ΔABC)=ar(ΔABD)
Two triangles are said to be on the common base and between the same parallels if
a) They have a common side.
b) The vertices opposite the common side lie on a straight line parallel to the common side.
Two Triangles Having the Common Base & Equal Areas
If two triangles have equal bases and are equal in area, then their corresponding altitudes are equal.
A Parallelogram and a Triangle Between the Same parallels
Theorem: If a triangle and a parallelogram are on the common base and between the same parallels, then the area of the triangle is equal to half the area of the parallelogram.
A triangle and a parallelogram are said to be on the same base and between the same parallels if
a) They have a common side.
b) The vertices opposite the common side lie on a straight line parallel to the common side. | 677.169 | 1 |
Given a point P on a triangular piece of paper A B C, consider the creases that are formed in the paper when A, B, and C are folded onto P. Let us call P a fold point of \triangle A B C if these creases, which number three unless P is one of the vertices, do not intersect. Suppose that A B=36, A C=72, and \angle B=90^{\circ}. Then the area of the set of all fold points of \triangle A B C can be written in the form q \pi-r \sqrt{s}, where q, r, and s are positive integers and s is not divisible by the square of any prime. What is q+r+s? | 677.169 | 1 |
Inequality
Triangles are some of the most basic yet fascinating geometric figures. While they appear simple, they abide by certain rules and theorems that ensure their integrity. One such rule is the Triangle Inequality Theorem. Imagine you're building a triangular fence for your garden. You have three wooden sticks of different lengths, but are they going […]
In math, we often talk about things being the same or equal, and that's where the properties of equality come in. They help us understand when and why numbers or expressions are equal. On the other hand, when we look at shapes and want to see if they're the same size and form, we talk […] | 677.169 | 1 |
are consecutive interior angles congruent
Have you ever wondered why some angles seem to fit perfectly together while others just don't? Well, that's where consecutive interior angles come into play! These seemingly simple angles hold a great deal of importance in the world of geometry and can help us solve complex problems. In this blog post, we'll dive into what consecutive interior angles are, their significance, and most importantly, whether or not they're congruent! So buckle up and get ready to unravel the mystery behind these fascinating angles.
What are consecutive interior angles?
Consecutive interior angles refer to a pair of angles that lie on the same side of a transversal and are located inside two parallel lines. To put it simply, they're the angles that sit next to each other within the interior space between two parallel lines. These types of angles might seem insignificant at first glance, but they play an important role in geometry. They help us solve problems related to parallel lines and transversals by providing us with valuable information about their measurements. For example, if we know that one consecutive angle measures 60 degrees, we can conclude that its adjacent angle will also measure 60 degrees since they're congruent by definition! This allows us to solve complex problems involving multiple angles and parallel lines. It's important to note that consecutive exterior angles are not congruent like their interior counterparts. Instead, these pairs of angles add up to form a straight line or 180 degrees. So when dealing with consecutive exterior angles, always remember this rule!
Why are they important?
Consecutive interior angles are important for several reasons. Firstly, they help us in understanding the properties of parallel lines and transversals. By knowing that consecutive interior angles are congruent, we can easily identify if two lines are parallel or not. Secondly, this property is used extensively in trigonometry to solve problems related to triangles. When we know the measure of one angle in a triangle, we can use the property of consecutive interior angles to find out other measures as well. Thirdly, by using this concept along with other geometric principles such as alternate interior angles and corresponding angles, it becomes easier to prove various geometric results like proving that a quadrilateral is a parallelogram or that two triangles are similar. Lastly but most importantly, understanding consecutive interior angles helps build a strong foundation for advanced mathematical concepts and theories. Whether you're studying geometry or calculus, having a solid grasp on basic principles like these will make it easier for you to understand more complex ideas down the line.
How to prove that consecutive interior angles are congruent
Proving that consecutive interior angles are congruent is not as difficult as it may seem. To start, let's first define what we mean by consecutive interior angles. These are the pairs of angles on the inside of two parallel lines that are crossed by a transversal. To prove that these consecutive interior angles are congruent, we can use the property known as alternate interior angles theorem. This states that when two parallel lines are crossed by a transversal, then the pairs of alternate interior angles formed will be equal in measure. Another way to prove this is through vertical angle theorem which states that if two intersecting lines form an X shape (also known as an "x" mark), then opposite or vertical angles formed from this intersection have equal measures regardless of their distances from each other. We can also use algebraic equations to prove that consecutive interior angles are congruent. By setting up a system of equations and solving for one variable using substitution or elimination method based on its corresponding equation with another variable set-up similarly but different value ranges given situational factors such as intersecting points within shapes or figures where they occur- one could easily show how these values end up being equivalent to each other upon solving them all together! There are various ways to prove that consecutive interior angles are congruent. Whether you choose to use geometry principles like alternate and vertical angle theorems or mathematical methods like algebraic equations – knowing how to do so will surely come in handy for any student studying math!
Examples of when consecutive interior angles are NOT congruent
Although most cases of consecutive interior angles are congruent, there are instances where they are not. One example is in a trapezoid, which is a quadrilateral with one pair of opposite sides parallel. In a trapezoid, the consecutive interior angles formed by the non-parallel sides have different measures. This can be proven using algebraic equations or geometric principles. Another example is in irregular polygons, where consecutive interior angles may vary depending on their position within the shape. The larger and more complex the polygon, the greater likelihood that some consecutive interior angles will not be congruent. Additionally, when lines intersect at an angle other than 90 degrees, such as in an acute or obtuse triangle or irregular quadrilateral, their consecutive interior angles will also differ in measure. While it's important to understand that most cases of consecutive interior angles are congruent, it's equally important to recognize when this rule does not apply and how to prove these exceptions using geometric principles and algebraic equations.
Conclusion
Understanding consecutive interior angles and their congruence is crucial in geometry. By definition, consecutive interior angles are the pairs of angles that are on opposite sides of the transversal but inside the two parallel lines. They have a special relationship: they are supplementary. Therefore, knowing that consecutive interior angles are congruent can help us prove various geometric theories and solve problems involving parallel lines and transversals. It also helps us to recognize patterns in different geometrical figures. Proving that these angles are congruent requires basic algebraic knowledge and an understanding of adjacent angle relationships. Once we establish this fact through rigorous proof methods such as using equations or logical arguments, it allows us to apply our knowledge to more complex problems efficiently. Recognizing the importance of this concept can lead to a deeper appreciation for geometry as well as making it easier for students to understand how shapes and objects relate mathematically. | 677.169 | 1 |
Synonyms for tangent
The term tangent is identified in base64 scheme by the sequence dGFuZ2VudA==, while the MD5 signature is equal to 75e7384f08d8f81a380699ce840c1167. The ASCII encoding of tangent in hexadecimal notation is 74616e67656e74. | 677.169 | 1 |
matMult
Parameters
unit
Calculate this point but as a unit vector from 0, 0, meaning
that the distance from the resulting point to the 0, 0
coordinate will be equal to 1 and the angle from the resulting
point to the 0, 0 coordinate will be the same as before. | 677.169 | 1 |
Finding the Scalar Triple Product of Three Vectors
In summary, to find the value of m that will result in a volume of 24 for a parallelopiped with edges given by (2î + 3j^+ 4k^), 4j^, and (5j^ + mk^), you can use the formula for the scalar triple product of three vectors. This formula is not the same as the dot product of three vectors.
Aug 30, 2018
#1
Ashutosh
3
2
Homework Statement
The edges of a parallelopiped are given by the vectors (2î + 3j^+ 4k^), 4j^ and (5j^ + mk^). What should be the value of m inorder that the volume of the parallelopiped be 24?
Homework Equations
The Attempt at a Solution
Volume of the parallelopiped is the scalar triple product of three vectors. I want to know the formula or method of finding dot product of three vectors.
The scalar triple product is not a dot product of three vectors, it is ##\vec u \cdot (\vec v \times \vec w)##.
LikesAshutosh
Aug 30, 2018
#3
Ashutosh
3
2
Orodruin said:
The scalar triple product is not a dot product of three vectors, it is ##\vec u \cdot (\vec v \times \vec w)##.
Thank You.
Related to Finding the Scalar Triple Product of Three Vectors
1. What is motion in a plane?
Motion in a plane refers to the movement of an object in two-dimensional space, along the x and y axes. This includes both linear and rotational motion.
2. How is motion in a plane different from motion in a straight line?
Motion in a plane involves movement along two axes, while motion in a straight line only involves movement along one axis. In other words, motion in a plane is two-dimensional, while motion in a straight line is one-dimensional.
3. What is the difference between velocity and speed in motion in a plane?
Velocity refers to the rate of change of an object's position in a specific direction, while speed refers to the rate of change of an object's position regardless of direction. In other words, velocity is a vector quantity, while speed is a scalar quantity.
4. How are vectors used in analyzing motion in a plane?
Vectors are used to represent both the magnitude and direction of an object's motion in a plane. They can also be used to calculate the resultant velocity and acceleration of an object.
5. Can motion in a plane be described using equations?
Yes, motion in a plane can be described using equations such as the equations of motion, which relate velocity, acceleration, time, and displacement. These equations can also be used to solve for unknowns in a motion in a plane problem. | 677.169 | 1 |
A polygon is any closed figure with sides made from straight lines. At each vertex of a polygon, there is both an interior and exterior angle, corresponding to the angles on the inside and outside of the closed figure. Understanding the relationships that govern these angles is useful in various geometrical problems. In particular, it is helpful to know how to calculate the sum of interior angles in a polygon. This can be done using a simple formula, or by dividing the polygon into triangles.
Method 1
Method 1 of 2:
Using the Formula
Download Article
1
Set up the formula for finding the sum of the interior angles. The formula is , where is the sum of the interior angles of the polygon, and equals the number of sides in the polygon.[1][2]
The value 180 comes from how many degrees are in a triangle. The other part of the formula, is a way to determine how many triangles the polygon can be divided into. So, essentially the formula is calculating the degrees inside the triangles that make up the polygon.[3]
This method will work whether you are working with a regular or irregular polygon. Regular and irregular polygons with the same number of sides will always have the same sum of interior angles, the difference only being that in a regular polygon, all interior angles have the same measurement.[4] In an irregular polygon, some of the angles will be smaller, some of the angles will be larger, but they will still add up to the same number of degrees that are in the regular shape.
2
Count the number of sides in your polygon. Remember that a polygon must have at least three straight sides.[5]
For example, if you want to know the sum of the interior angles of a hexagon, you would count 6 sides.
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3
Plug the value of into the formula. Remember, is the number of sides in your polygon.[6]
For example, if you are working with a hexagon, , since a hexagon has 6 sides. So, your formula should look like this:
4
Solve for . To do this, subtract 2 from the number of sides, and multiply the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon.[7]
For example, to find out the sum of the interior angles of a hexagon, you would calculate: So, the sum of the interior angles of a hexagon is 720 degrees.
Drawing Triangles
Draw the polygon whose angles you need to sum. The polygon can have any number of sides and can be regular or irregular.
For example, you might want to find the sum of the interior angles of a hexagon, so you would draw a six-sided shape.
2
Choose one vertex. Label this vertex A.
A vertex is a point where two sides of a polygon meet.
3
Draw a straight line from Point A to each other vertex in the polygon. The lines should not cross. You should create a number of triangles.
You do not have to draw lines to the adjacent vertices, since they are already connected by a side.
For example, for a hexagon you should draw three lines, dividing the shape into 4 triangles.
4
Multiply the number of triangles you created by 180. Since there are 180 degrees in a triangle, by multiplying the number of triangles in your polygon by 180, you can find the sum of the interior angles of your polygon.[8]
For example, since you divided your hexagon into 4 triangles, you would calculate to find a total of 720 degrees in the interior of your polygon.
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Community Q&A
Work out what all the interior adds up to, then divide by however many sides the shape has 34Helpful 46
Question
How do I calculate the number of sides of a polygon if the sum of the interior angles is 1080? 38
Question
If two equilateral triangles are placed together to form a rhombus, how do I calculate the value of each interior angle of this rhombus, and how do I find the sum?
Donagan
Top Answerer
In the rhombus you describe, the two smaller interior angles would each be 60°, and the two larger interior angles would each be 120°. You wouldn't have to calculate the angles. Simple inspection of the rhombus and the two triangles would show what the angles are, given that equilateral triangles have three 60° angles. The sum is 60° + 60° + 120° + 120°Check your work on a piece of paper using a protractor to sum the interior angles manually. When doing this, be careful while drawing the polygon's sides as they should be linear.
Thanks
Helpful1Not Helpful2Protractor (optional)
Pen
Eraser
Ruler
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About This Article
Co-authored by:
David Jia
Academic Tutor 423,591 times.
12 votes - 76%
Co-authors: 17
Updated: October 3, 2022
Views:423,591
Categories: Geometry
Article SummaryX
To calculate the sum of interior angles, start by counting the number of sides in your polygon. Next, plug this number into the formula for the "n" value. Then, solve for "n" by subtracting 2 from the number of sides and multiplying the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon! To learn how to calculate the sum of interior angles by drawing triangles, read on!
Did this article help you?
FAQs angle between two sides of a triangle is called the interior angle. ItSubtract each angle from 180 degrees to get the internal angles, you see that the answer is 8 x 180 - 360 = 1080 degrees. Check this for a regular octagon, in which the internal angles are 135 degrees. You see that 8 x 135 = 1080, so it checks out.
Octagon is a polygon with 8 sides. For an n-sided polygon, the formula to calculate the sum of interior angles is given by (n - 2) × 180°. So, since the octagon has 8 sides, the sum of the interior angles will be (8 - 2) × 180° = 1080°.
We also know that a rectangle (or any quadrilateral for that matter) has 4 sides: This means that the sum of all the internal angles of a rectangle is 180° × (4 - 2) = 180° × 2 = 360°. Hence, the sum of all the internal angles of a rectangle (or any quadrilateral) is 360°. | 677.169 | 1 |
Are some parallelogram a square?
All squares are parallelograms. The definition of parallelogram
is a quadrilateral with two sets of parallel sides. A square is a
quadrilateral with two sets of parallel lines, so it is also a
parallelogram | 677.169 | 1 |
Which side is converging in function? / right Which side could be used as a make-up mirror? / right Which side could be used as a side mirror in a car? / left Which point is closest to focus? / 3
b.
an upright magnified image? / 2 an inverted unmagnified image? / 5 an inverted image that is smaller than the object? / 6 no image at all / 3 an upright image that is smaller than the object? / 1 an inverted image that is larger than the object? / 4
52. a.
b.
Virtual, it forms on the same side as the object.
c.
1/do + 1/di = 1/f 1/6 + 1/di = 1/10 \ di = -15 cm
d.
hi/ho = -di/do = -(-15)/6 = 2.5
e.
1/do + 1/di = 1/f 1/20 + 1/di = 1/10 \ di = 20 cm (M = -di/do = -20/20 = -1) The image is on the left side at 20 cm. The image is inverted and the same size as the object. | 677.169 | 1 |
CBSE Sample Papers for Class 9 Maths Paper 4 4 of Solved CBSE Sample Papers for Class 9 Maths is given below with free PDF download solutions.
Question 16. Prove that the perimeter of a triangle is greater than the sum of their medians.
Question 17. In the given figure, m and n are two mirrors placed parallel to each other. An incident ray AB strikes the mirror m at point B and then reflected to the mirror n along path BC and again reflects back along CD. Prove that AB || CD.
Question 18. Prove that the quadrilateral formed by the internal angle bisectors of any quadrilateral is cyclic.
Question 19. Construct a right triangle whose base is 12 cm and sum of its hypotenuse and their side is 18 cm. Also verify it.
Question 20. An umbrella is made by stitching 10 triangular pieces of cloth of two different colours. Each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?
Question 21. The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the metal weighs 8.9 gm per cm3?
Question 22. A fair die is thrown 120 times with the following frequencies of number divisible by 3 and not divisible by 3. Divisible by 3 : 56 Not divisible by 3 : 64 Find the probability when the number is (i) divisible by 3 (ii) Not divisible by 3.
Question 25. Seema wants to invest Rs 20000 in two types of bond. She earns 12% on the first type and 15% on the second type. Find the investment in each if her total earning is Rs 2850. Find how much she invests at each type of bond. Write two values which are depicted here.
Question 26. In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that (i) ∆ AMC = ∆ BMD (ii) ∠DBC is a right angle or ∠DBC = ∠ACB = 90° (iii) ∆DCB ≅ ∆ACB (iv) CM = \(\frac { 1 }{ 2 }\)AB
Question 27. XY is a line parallel to side BC of a ∆ABC. If BE || AC and E and F respectively. Show that ar (∆ABE) = ar (∆ACF).
Question 28. A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone. OR There is one and only one circle passing through three given non-collinear points.
Question 29. A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and diameter of the graphite is 1 mm. If the length of the pencil is 14 m. Find the (i) Volume of graphite (ii) Weight of graphite (iii) Volume of wood (iv) Weight of pencil. If density of wood = 0.7 gm/cm3 and density of graphite = 2.1 gm/cm3.
Question 30. If n observations x1, x2, x3 …, xn have sum of deviations -10 from 50 and sum of deviations 70 from 46, then find the value of n and mean of the observations.
Solution 25. Let Seema invest Rs x on first type of bonds and Rs (20000 – x) on the second type. Investment on first type of bonds = Rs 5000 Investment on second type of bonds = Rs (20000 – 5000) = Rs 15,000. Two values depicted here are (i) Saving habit (ii) Care for future (iii) Help the country for development
Solution 27. Given: A ∆ABC in which XY || BC, BE || CA and CF || BA. => BE || CY and CF || BX To prove: ar (∆ABE) = ar (∆ACF) Proof: ||gm EBCY and ∆ABE are on the same base BE and between the same parallels BE & CA. ar (∆ABE) = \(\frac { 1 }{ 2 }\) ar (||gm EBCY) …(1) Again ||gm BCFX and ∆ACF are on the same base CF and between the same parallels CF and BA. => ar (∆ACF) = \(\frac { 1 }{ 2 }\) ar (||gm BCFX) …(2) But ||gm EBCY and ||gm BCFX are on the same base BC and between the same parallels BC and EF. => ar (||gm EBCY) = ar (||gm BCFX) …(3) From (1), (2) and (3) ar(∆ABE) = ar(∆ACF)
Solution 28. Given: Three non-collinear points P, Q and R. To prove: A circle passes through these three points P, Q and R, and such circle is one and only one. Construction: Join PQ and QR. Their perpendicular bisectors AL and BM intersect at O. Join OP, OQ and OR. Proof: ∵ Point O is at the perpendicular bisector of chord PQ. ∴ OP = OQ …(i) Similarly O is at the perpendicular bisector of chord QR => OQ = OR …(ii) From Eqn. (i) & (ii) OP = OQ = OR = r (Let) Now taking O as a centre and r radius if we draw a circle it will pass through all three points P, Q and R, i.e., P, Q and R exist on the circumference of the circle. Now let another circle be (O', s) which pass through points P, Q and R and perpendicular bisectors of PQ and QR i.e., AL and BM passes through the centre O'. But the intersection point of AL and BM is O. i.e. O' and O coincide each other or O and O' are the same point ∴ OP = r and OP' = s and O and O' are coincide => r = s. => C (O, r) = C (O', s) => There is one and only one circle through which three non-collinear points P, Q and R pass. => There is one and only one circle passing through three given non-collinear points.
We hope the CBSE Sample Papers for Class 9 Maths Paper 4 help you. If you have any query regarding CBSE Sample Papers for Class 9 Maths Paper 4, drop a comment below and we will get back to you at the earliest. | 677.169 | 1 |
Today while using the Geogebra program I discovered a new engineering feature, I don't know if it was already known or not
Data: $R$ is a circle with center $O$, $A,B,C,D∈R$, $AD∩BC=E$, $AD∩BC=F$, $R_1$ is the passing circle passing through the points $A,B,E$, its center is $M$, and $R_2$ is the passing circle passing through the points $A,C,D$, its center is $N$, $R_3$ is the passing circle passing through the points $B,C,F$, its center is $P$, and $R_4$ is the passing circle passing through the points $A,D,F$, its center is $Q$, and $R_1∩R_2∩R_3∩R_4=S$.
Required: Prove that the points $M,N,O,P,Q,S$ lie on one circle.
But I don't know how to prove it.
Whoever can help, please kindly
If the feature is already known, please mention a source that talks about it, and thank you
$\begingroup$Are you sure there is no additional condition in your case? In the link you gave it seems the center of the initial triangle (called O in your case) would not be cocyclic with the other centers. Maybe BCD isocele?$\endgroup$
1 Answer
1
From a point outside the given circle $R$ we draw two lines that determine the points $A,B,C,D$ of the circle which in turn determines the point $E$.
The center $Q$ of $R_4$ is the intersection of the bisectors of the sides $AF$ and $FD$ of the triangle $\triangle{AFD}$ and the center $P$ of $R_3$ is the intersection of the bisectors of the sides $BF$ and $FC$ of the triangle $\triangle{BFC}$.
The points $O,P,Q$ determine an unique circle (which will be the red circle of the statement).
A procedure exactly the same as the one we have just given determines the centers $M$ of $R_1$ and $N$ of $R_2$ and then we can optionally verify that $M$ and $N$ belong to the red circle or see that the equation of the circumcircle of the triangle $\triangle{OMN}$ coincides with the preceding one of the triangle $\triangle{OPQ}$. | 677.169 | 1 |
Unit Circle Complete
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Unit Circle Labeled With Special Angles And Values ClipArt ETC
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Web A Unit Circle Is Formed With Its Center At The Point (0, 0), Which Is The Origin Of The Coordinate Axes.
Web you might be given a complete unit circle, with the values for the angles in the other three quadrants, too. Solution note11negative angles on a number line, a positive. Sine, cosine and tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled. It has all of the angles in radians and degrees.
Web Welcome To 8813 Stone Ridge Circle, Unit #301!
Like many ideas in math, its simplicity makes it beautiful. Web select the unit number from the dropdown. Web draw the unit circle. Web nearby recently sold homes.
Web Check Out This Amazing Property Located Right In The Heart Of Rosedale.
Web what is a unit circle chart? [1] frequently, especially in trigonometry, the. Web in mathematics, a unit circle is a circle of unit radius —that is, a radius of 1. Generate every radian measure just by counting.
Web For Each Point On The Unit Circle, Select The Angle That Corresponds To It.
Here you can download a copy of the unit circle. Get resident's name, phone, email and social media in seconds! 0 ∘ ≤ α < 360 ∘. Nearby homes similar to 5205 wyndholme cir #104 have recently sold between $109k to $400k at. | 677.169 | 1 |
Videos in this series
Please select a video from the same chapter
This video is the first in a series on Geometry. This one looks at Angles and Triangles which is part of the Year 9 course. Designed as an recap of the work which has been covered in previous years, it's a short video filled with really important information. This will form the foundation for the work which is to come. There are lots of | 677.169 | 1 |
How to convert radians to degrees calculator
Converting radians to degrees is a common task in various fields, such as mathematics, physics, and engineering. This article will provide you with detailed steps on how to use a calculator for this conversion, guiding you through the process easily and accurately.
Step 1: Understand the relationship between radians and degrees
Before diving into the calculations, it's essential to know the basic relationship between radians and degrees. One complete circle (360°) equals 2π (pi) radians. Therefore, the formula to convert radians (rad) to degrees (°) is:
Degrees = Radians × 180 / π
Step 2: Select an appropriate calculator
Choose a calculator that offers support for both degree and radian mode, as well as access to the mathematical constant π (pi). Scientific calculators or online tools, such as Google's built-in calculator or Wolfram Alpha, are usually suitable options.
Step 3: Input the radian value
Enter the given radian value you wish to convert into the calculator. Make sure your calculator is set in 'Radian mode' if it offers the option; otherwise, input the radian value directly.
Step 4: Apply the conversion formula
Multiply the input radian value by 180. If your calculator has a designated π button, divide the result by π. If not, divide by approximately 3.14159.
Step 5: Obtain the result in degrees
After performing these calculations, your calculator will display the converted angle in degrees. Some calculators may present automatic formatting or notation for angles in degree format; otherwise, make a note of the result as being in degrees.
Step 6: Verify your calculations
To ensure accuracy and double-check your work, try entering your calculated degree value back into your calculator and converting it back into radians using this formula:
Radians = Degrees × π / 180
If you obtain the original radian value, you may trust the accuracy of your conversion.
Conclusion:
Being proficient in converting radians to degrees using a calculator is not only remarkably useful but also quite simple. Just be mindful of the underlying relationship between radians and degrees, select an appropriate calculator, and follow the necessary steps. With practice, you'll become extremely comfortable with these conversions | 677.169 | 1 |
Let ABABAB and CDCDCD be the two temples and ACACAC be the river. Let the height of temple ABABAB be
505050 m. ACACAC is the river. The angles are depression are shown as corresponding angles of elevation. Let the
height of CDCDCD be xxx m and width of river be www m. Thus, CD=xCD = xCD=x and AC=wAC = wAC=w.
Let BEBEBE be the tower leaning due east where BBB is the foot of the tower and EEE is the top. ABABAB is
the vertical height of the tower taken as hhh. The angles of elevation are shown from tow points as given in the
question.
Let ACACAC be the lake and BBB be the point of observation 250025002500 m above lake. Let EEE be the cloud and
FFF be its reflection in the lake. If we take height of the cloud above lake as hhh then CD=2500CD = 2500CD=2500 m where
BD∣∣ACBD||ACBD∣∣AC . DE=h−2500DE = h - 2500DE=h−2500 m and CD=2500CD = 2500CD=2500 m. The angle of elevation and angle of depression of cloud and
its reflection are shown as given in the problem.
This is a problem similar to previous problem with 250025002500 replaced by hhh and angles are replaced by
α\alphaα and β\betaβ. So the diagram is similar in nature. Let the height of the cloud above lake be
h′h'h′ m. So DE=h′−hDE = h' - hDE=h′−h and DF=h+h′DF = h + h'DF=h+h′.
Also, secα=BEBD⇒BE=2hsecαtanα−tanβ\sec\alpha = \frac{BE}{BD} \Rightarrow BE = \frac{2h\sec\alpha}{\tan\alpha - \tan\beta}secα=BDBE⇒BE=tanα−tanβ2hsecα which is the distance of
the cloud from the point of observation.
The diagram is given below:
Let ABABAB be the height of plane above horizontal ground as hhh miles. CCC and DDD are two consecutive
milestones so CD=1CD = 1CD=1 mile. Let BC=xBC = xBC=x mile. The angles of depression are represented as angles of elevation.
Let PQPQPQ be the post with height hhh and ABABAB be the tower. Given that the angles of elevation of BBB
at PPP and QQQ are α\alphaα and β\betaβ respectively. Draw CQ∣∣PACQ||PACQ∣∣PA such that PQ=AC=hPQ = AC =
hPQ=AC=h and A:math:AP = QC = x. Also, let BC=h′BC = h'BC=h′ so that AB=AC+BC=h+h′AB = AC + BC = h + h'AB=AC+BC=h+h′.
Let ADADAD be the wall, BDBDBD and CECECE are two positions of the ladder. Then according to question BC=a,DE=bBC =
a, DE = bBC=a,DE=b and angles of elevations at BBB and CCC are α\alphaα and β\betaβ. Let AB=xAB = xAB=x and
AE=yAE = yAE=y. Also, let length of ladder be lll i.e BD=CE=lBD = CE = lBD=CE=l.
Let CDCDCD be the tower subtending angle α\alphaα at AAA. Let BBB be the point bbb m above
AAA from which angle of depression to foot of tower at CCC is β\betaβ which is shown as angle of elevation.
Let AC=xAC = xAC=x and CD=hCD = hCD=h.
Let ABABAB be the observer with a height of 1.51.51.5 m, 28.528.528.5 m i.e. ADADAD from tower DE,30DE, 30DE,30 m
high. Draw BC∣∣ADBC||ADBC∣∣AD such that AB=CD=1.5AB = CD = 1.5AB=CD=1.5 m and thus CE=28.5CE = 28.5CE=28.5 m. Let the angle of elevation from
observer's eye to the top of the tower be α\alphaα.
Let ABABAB be the tower havin a height of hhh and CCC and DDD are two objects at a distance of xxx
and x+yx + yx+y such that angles of depression shown as angles of elevatin are β\betaβ and α\alphaα respectively.
Let ABABAB be the height of the window at a height hhh and DEDEDE be the house opposite to it. Let the distance
between the houses be AD=xAD = xAD=x. Draw BC∣∣ADBC||ADBC∣∣AD such that BC=xBC = xBC=x and CD=hCD = hCD=h. The angles are shown as
given in the problem. Let CE=yCE = yCE=y
Total height of the second house DE=CD+DE=y+h=h(1+tanαcotβ)DE = CD + DE = y + h = h(1 + \tan\alpha\cot\beta)DE=CD+DE=y+h=h(1+tanαcotβ)
The diagram is given below:
Let ADADAD be the ground, BBB be the lower window at a height of 222 m, CCC be the upper window at a
height of 444 m above lower window and GGG be the balloon at a height of x+2+4x + 2 + 4x+2+4 m above ground. Draw
DG∣∣AC,BE∣∣ADDG||AC, BE||ADDG∣∣AC,BE∣∣AD and CF∣∣ADCF||ADCF∣∣AD so that DE=2DE = 2DE=2 m, EF=4EF = 4EF=4 m and FG=xFG = xFG=x m. Also, let BE=CF=dBE
= CF = dBE=CF=d m. The angles of elevation are shown as given in the problem.
Let ABABAB be the lamp post, EFEFEF and GHGHGH be the two positions of the man having
height 666 ft. Let the shdows be ECECEC and GDGDGD of lengths 242424 ft. and
303030 ft. for initial and final position. Since the man moves eastward from his initial position
∴∠ACD=90∘\therefore \angle ACD = 90^\circ∴∠ACD=90∘.
Let ABABAB be the tower having a height of hhh m, ACACAC be the final length of shadow taken as xxx m,
ADADAD is the initial length of shadow which is 555 m more than finla length i.e. CD=5CD = 5CD=5 m. The angles of
elevation are shown as given in the problem.
Let AAA be the initial position of the man and DDD and EEE be the objects in the west. Let DE=x,AD=y,∠ADB=θ,∠AEB=ϕDE = x, AD
= y, \angle ADB=\theta, \angle AEB = \phiDE=x,AD=y,∠ADB=θ,∠AEB=ϕ and ∠ADC=ψ\angle ADC=\psi∠ADC=ψ. α\alphaα and β\betaβ are the angles made
by objects on the two positions of the man as given in the problem.
Let PPP be the object and OAOAOA be the straight line on which BBB and CCC lie underneath the object.
Let OP=hOP = hOP=h. According to question the angles of elevation made are α,2α\alpha, 2\alphaα,2α and 3α3\alpha3α from
A,BA, BA,B and CCC i.e. ∠PCO=3α,∠PBO=2α\angle PCO = 3\alpha, \angle PBO = 2\alpha∠PCO=3α,∠PBO=2α and ∠PAO=α\angle PAO = \alpha∠PAO=α. Given that
AB=αAB = \alphaAB=α and BC=bBC = bBC=b.
Given that AAA and BBB are two points of observation on ground 100010001000 m apart. Let
CCC be the point where the balloon will hit the ground at a distance xxx m from
BBB. Also, let DDD and EEE be the points above AAA and BBB respectively
such that ∠BAE=30∘\angle BAE= 30^\circ∠BAE=30∘ and ∠DBA=60∘\angle DBA = 60^\circ∠DBA=60∘.
Let ABABAB be tree having height hhh m and BCBCBC be the width of the river having width
www m. According to question angle of elevation of the tree from the opposite bank is
60∘60^\circ60∘. Also, let DDD be the point when the man retires 404040 m from where the
angle of elevation of the tree is 30∘30^\circ30∘.
Thus, width of the river is 202020 m and height of the tree is 20320\sqrt{3}203 m.
The diagram is given below:
Let OOO be the point of observation. The bird is flying in the horizontal line WXYZWXYZWXYZ. The
angles of elevation of the bird is given at equal intervals of time. Since the speed of the bird is
constant WX=XY=YZ=yWX = XY = YZ = yWX=XY=YZ=y (let). From question ∠AOW=α,∠BOX=β,∠COY=γ\angle AOW = \alpha, \angle BOX = \beta,
\angle COY = \gamma∠AOW=α,∠BOX=β,∠COY=γ and ∠DOZ=δ\angle DOZ = \delta∠DOZ=δ. Let OA=xOA = xOA=x and AW=hAW = hAW=h.
Let ABABAB be the tower, BCBCBC be the pole and DDD be the point of observation where the
tower and the pole make angles α\alphaα and β\betaβ respectively. Let the height of the
tower be h′h'h′ and AD=dAD = dAD=d. Given that the height of the pole is hhh.
Let ABABAB be the first chimney and CDCDCD be the second chimney. The angles of elevation are
shown as angles of elevation as given in the problem. Draw BE∣∣ACBE||ACBE∣∣AC and let AC=BE=dAC = BE = dAC=BE=d m
and AB=CE=hAB = CE = hAB=CE=h m. Given CD=150CD = 150CD=150 m. Clearly, DE=150−hDE = 150 - hDE=150−h m.
Let CDCDCD be the tower of height hhh having an elevation of 30∘30^\circ30∘ from AAA
which is southward of it. Let BBB be eastward of AAA at a distance of aaa from it
from where the angle of elevation is 18∘18^\circ18∘. Since BBB is eastward of A∠CAB=90∘A
\angle CAB = 90^\circA∠CAB=90∘.
Let BBB be the peak having a height of hhh with base AAA. Let PQPQPQ is the
horizontal base having a length 2a2a2a making angle of elevation of θ\thetaθ from each
end. Let RRR be the mid-point of PQPQPQ from where the angle of elevation of BBB is
ϕ\phiϕ as given in the question.
Let BBB be the top of the hill such that height of the hill ABABAB is hhh and P,R,QP,
R, QP,R,Q be the three consecutive milestones. Given, ∠APB=α,∠ARB=β,∠AQB=γ\angle APB = \alpha, \angle ARB = \beta,
\angle AQB = \gamma∠APB=α,∠ARB=β,∠AQB=γ.
Let OPOPOP be the tower haing a height of hhh which is to be found. Let ABCABCABC be the
equilateral triangle. Given that OPOPOP subtends angles of α,β,γ\alpha, \beta, \gammaα,β,γ at
A,B,CA, B, CA,B,C respectively. Given that tanα=3+1\tan\alpha = \sqrt{3} + 1tanα=3+1 and tanβ=tanγ=2\tan\beta =
\tan\gamma = \sqrt{2}tanβ=tanγ=2. It is given that OPOPOP is perpendicular to the plane of △ABC\triangle
ABC△ABC.
In the diagram we have shown only one tower instead of three. We will apply cyclic formula to this one
tower relationships. Let PPP be the position of the eye and height of PQ=xPQ = xPQ=x. Let
ABABAB be the tower having a height of aaa as given in the question and let the angle
subtended by ABABAB at PPP is θ\thetaθ.
Let SSS be the initial position of the man and PPP and QQQ be the poosition of the
objects. Since PQPQPQ subtends greatest angle at RRR, a circle will pass through P,QP, QP,Q
and RRR and RSRSRS will be a tangent to this circle at RRR.
Let OPOPOP be the tower having a height of hhh and PQPQPQ be the flag-staff having a
height of xxx. AAA and BBB are the two points on the horizontal line OAOAOA. Let
OB=yOB = yOB=y. Given, AB=d,∠QAP=∠QBP=αAB = d, \angle QAP = \angle QBP = \alphaAB=d,∠QAP=∠QBP=α.
Since ∠QAP=∠QBP\angle QAP = \angle QBP∠QAP=∠QBP, a circle will pass through the points A,B,PA, B, PA,B,P and
QQQ because angles in the same segment of a circle are equal.
This problem is similar to 138138138 and has been left as an exercise.
This problem is similar to 969696 and has been left as an exercise. The answer is 909090 seconds.
The diagram is given below:
Let CCC be the position of the aeroplane flying 300030003000 m above ground and DDD be the
aeroplane below it. Given that the angles of elevation of these aeroplanes are 45∘45^\circ45∘ and
60∘60^\circ60∘ respectively. Let the height of DDD is hhh m and AB=dAB = dAB=d m.
Let CCC and DDD be two consecutive milestones so that CD=1CD = 1CD=1 mile. Let DDD be
position of aeroplane having a height hhh above AAA, to which angles of elevation are
α\alphaα and β\betaβ from CCC and DDD respectively. Let AC=x⇒AD=1−xAC = x
\Rightarrow AD = 1 - xAC=x⇒AD=1−x.
This problem is simmilar to 138138138, and has been left as an exercise.
The diagram is given below:
Let ABABAB be the height of air-pilot which has height of hhh. Let CDCDCD be the tower
whose angles of depression of top and bottom of tower be 30∘30^\circ30∘ and 60∘60^\circ60∘
respectively. Draw DE∣∣ACDE||ACDE∣∣AC such that DE=ACDE = ACDE=AC. Let the height of tower CDCDCD be
xxx. | 677.169 | 1 |
Elements of Geometry: Containing the First Six Books of Euclid with a Supplement on the Quadrature of the Circle, and the Geometry of Solids : to which are Added, 60.
Óĺëßäá 9 ... diameter of a circle is a straight line drawn through the centre , and terminated both ways by the circumference . 14. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter . 15 ...
Óĺëßäá 33 ... diameter is a straight line joining two of its opposite angles . Let ACDB be a parallelogram , of which BC is a diameter ; the oppo- site sides and angles of the figure are equal to one another ; and the diam- eter BC bisects it . A B D ...
Óĺëßäá 35 ... diameter AB bisects ( 34. 1. ) it ; and the triangle DBC is the half of the parallelogram DBCF , because the diameter DC bisects it ; and the halves of equal things are equal ( 7 . Ax . ) ; therefore the triangle ABC is equal to the ...
Óĺëßäá 37 ... diameter of any parallelogram , are equal to one another . Let ABCD be a parallelogram of which the diameter is AC ; let EH , FG be the parallelograms about AC , that is , through which AC passes , and let BK , KD be the other ...
ÄçěďöéëŢ áđďóđÜóěáôá
Óĺëßäá 41 70 61 - THE diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote ; and the greater is nearer to the centre than the less.* Let ABCD be a circle, of which... with a Supplement on the Quadrature of the Circle, and the Geometry of Solids : to which are Added, Elements of Plane and Spherical Trigonometry | 677.169 | 1 |
A chord of a circle equal to m subtracts an arc of 120 °. Find the radius of the circle.
Since the chord AB contracts the arc at 120, the central angle resting on this arc is also equal to 120.
From point O, the center of the circle, draw the radii of the circle OA and OB.
Since ОА = ОВ = R, the triangle AOB is isosceles. Let's draw the height OH to the base AB of the triangle ABC. In an isosceles triangle, the height is also the bisector and median of the triangle. Then AH = BH = m / 2, and the angle AOH = BOH = 120/2 = 60.
In a right-angled triangle АHО SinАН = АH / ОА.
OA = AH / Sin60.
ОА = (m / 2) / (√3 / 2) = m / | 677.169 | 1 |
You are given a convex polygon, ie all its internal angles are less than 180 degrees. Prove that you can always draw three straight lines through a specific point inside this polygon, such that they divide it into 6 equal (by area) regions? Bonus questions:
Can you prove that this can be achieved for non-convex polygons too? This time the point may not lie inside the polygon.
Does this result extend to 4 lines dividing a convex polygon into 8 equal regions?
This question was inspired by the hard problem in the recent TopCoder Open Algorithm final, written by Michal Forisek (misof).
1 Answer
1
Let's have a polygon. Let's draw a line at some angle $\theta_1$ to some fixed direction. By moving the line parallel to itself, we can make the whole polygon to lie either on one side of the line or the other. Thus, since the function "area on one side minus area on the another" is continuous, it should go through zero. Thus, for every $\theta_1$ there is a line $AD = L(\theta_1)$ that splits the polygon in halves.
Using the same mean value theorem we can show, that for every $\theta_1$, there is $\theta_2$ such that lines $L_1=AD=L(\theta_1)$ and $L_2=BE=L(\theta_2)$ split the area in a proportion $k=[APB]/[BPD]=1/2$. Indeed, if $\theta_2=\theta_1$, then $L_1=L_2$ and $k=0$. However, if $\theta_2=\theta_1+\pi$ (we rotated the line 2 by $\pi$ and $L_1=L_2$ again), then $k=\infty$. Given $k(\theta_2)$ is continuous, there should be some $\theta_2$, so $k=1/2$
By the same argument, for every $\theta_1$ there is always a $\theta_3$ and line $L_3=CF=L(\theta_3)$, so $k=[AQC]/[CQD]=2$.
Now consider point $R$, the intersection of $L_2$ and $L_3$. As we traverse from $\theta_1\to\theta_1+\pi$, lines $L2\leftrightarrow L3$ and $P\leftrightarrow Q$, but $R\to R$. However, now $R$ lies on the other side of line $L_1$. That means that during its journey, it crossed the line $L_1$. At this moment all three lines pass through one point.
By the way, we didn't use neither property of convexity, nor that it is a polygon. What mattered is that the area of the intersection of this shape with half-plane is continuous in respect to movements of the half-plane.
$\begingroup$So how can we ensure the 'jorney' mentioned in paragraph 4 is continuous?$\endgroup$
– user376921
CommentedNov 21, 2019 at 12:29
$\begingroup$It takes several formalities, but it boils down to showing: 1) $L(\theta)$ is continuous (e.g, in terms of intersections with large circle); 2) $\theta_{2,3}(\theta_1)$ are continuous; 3) lines $L_2$ and $L_3$ are never coincident or parallel => $R(\theta_1)$ is a continuous motion$\endgroup$
$\begingroup$In my understnading, 1) distance between $L(\theta)$ is taken to be supremum of distance between lines inside the polygon. 2) $\theta _2,\theta_3$ as output, are well-defined due to continuity of L under the metric, continuous since intuition (if $L$ close to $L(\theta_1)$, then $\theta_{2,3}(\theta_1) $ close to $ \theta_{2,3}(L)$ ). 3) $\theta _1 \rightarrow \theta _1 +\pi$ is just a flip in orientation, and $R(\theta _1) $ continuous as intuition. To add, since $ L(\theta)$ continuous as shown, the triplet $ (P,Q,R)$ is then continuous. So this triplet have equal entries in some $\theta$$\endgroup$ | 677.169 | 1 |
2. E and F are points on the sides PQ and PR, respectively of a ΔPQR. For each of the following cases, state whether EF || QR. (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 3.
3. In the figure, if LM || CB and LN || CD, prove that AM/AB = AN/AD
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 4.
4. In the figure, DE||AC and DF||AE. Prove that BF/FE = BE/EC
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 5.
5. In the figure, DE||OQ and DF||OR, show that EF||QR.
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 6.
6. In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 7.
7. Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 8.
8. Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 9.
9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO/BO = CO/DO.
Solution:
Class 10 Maths 6.2 NCERT Solutions Question 10.
10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/DO. Show that ABCD is a trapezium.
Solution:
Maths in Class 10: Triangles
In Class 10 Maths 6 Triangles.
Basic Proportionality Theorem (BPT) and its Converse
Basic Proportionality Theorem If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then other two sides are divided in the same ratio. Thus in ∆ABC, if DE || BC, then
Converse of BPT If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side, 6 Triangles.
What is the Advantage of Class 10 Maths NCERT Solutions Chapter 6 Provided by Bhautik Study?
Class 10 Maths 6.2 NCERT Solutions
In mathematics, a triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. In class 10, students typically learn about various properties and concepts related to triangles. Here's a brief overview of what is usually covered in Class 10 Maths 6.2 NCERT Solutions:
Types of Triangles:
Equilateral Triangle: All three sides are of equal length, and all three angles are equal (each measuring 60 degrees).
Isosceles Triangle: Two sides are of equal length, and consequently, two angles are equal.
Scalene Triangle: No sides are of equal length, and no angles are equal.
Angles:
Sum of Angles: The sum of the interior angles of a triangle is always 180 degrees.
Exterior Angle: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Properties:
Side Lengths: The sum of the lengths of any two sides of a triangle is greater than the length of the third side (Triangle Inequality Theorem).
Altitudes: Perpendiculars drawn from each vertex to the opposite side are called altitudes. The point where the altitude intersects the side is called the foot of the altitude.
Medians: A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
Angle Bisectors: An angle bisector of a triangle is a line segment that bisects one of the vertex angles of the triangle.
Orthocenter: The point where all three altitudes of a triangle intersect.
Centroid: The point where all three medians of a triangle intersect.
Circumcenter: The point where the perpendicular bisectors of the sides of a triangle intersect.
Congruence and Similarity:
Triangles are congruent if their corresponding sides and angles are equal.
Triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.
Pythagoras Theorem:
In a right-angled 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.
Area of a Triangle:
Various methods can be used to calculate the area of a triangle, including the formula: Area = (1/2) * base * height.
These are some fundamental concepts related to triangles that are typically covered in class 10 mathematics curriculum. Students often solve problems and apply these concepts to various geometric and real-world situations here Class 10 Maths 6.2 NCERT Solutions play an important role. | 677.169 | 1 |
distance between A and B
6
7
0
8
Hint:
In this question we have to count the total number of points between point A and B. In the given figure the distance between two consecutive point is unit.
The correct answer is: 6
In this question we have to count the total number of points between point A and B. In the given figure the distance between two consecutive point is unit. From the figure we can easily see that there are points. So, the distance between points A and is | 677.169 | 1 |
At least two views of orthographic projection are selected. Web how to draw objects in oblique projection 📦 cabinet and cavalier drawings arthur geometry 79.8k subscribers subscribe 40 share 2k views 4 months ago #howtodraw #geometry learn what the. Web thus, cabinet projection is the preferred method for constructing an oblique projection. 45 degrees is the angle for all lines drawn backwards. Web in this video, i'll teach you all you need to know about oblique projection. Draw the front or side view of the object. There are three types of pictorial views:
Oblique Drawing, Projection its Types, Examples. CivilSeek
Web to draw it in oblique projection follow three main rules: Web after clicking on the button below, you will find a file in.pdf format in the downloads folder on your computer containing the free.
What is Oblique Projection Types Cavalier, and General
There are three types of pictorial views: Then draw the back face of the cube measuring 4 x 4 a little further from the first cube, which will offset with. Let's make an oblique drawing.
Emmanuel Garcia I like to draw this way Projection systems
Then all the three axes of the oblique drawing are constructed in which one axis is horizontal, and the other is. Web oblique projection is a type of parallel projection: 45 degrees is the angle.
What Is Oblique Drawing? Oblique Projection Oblique Drawing
Web to draw it in oblique projection follow three main rules: An oblique drawing of the bearing bracket in cabinet projection is shown in figure 2.8. First draw the front face of the cube measuring.
How To Draw An Oblique Drawing
How to draw the oblique proportion perspective, by mattias adolfsson. Start by drawing a few basic shapes to represent the object. Web oblique drawing is a drawing in which the front view of the object.
How To Draw An Oblique Drawing
Web in this video, i teach you all you need to know about oblique projection. Web after clicking on the button below, you will find a file in.pdf format in the downloads folder on your.
Oblique projection example 8
I'll cover all the basics of oblique drawing for engineering and technical drawing students. Next, locate all given points on any given block according to the dimensions given. Web in this video, i teach you.
How To Draw An Oblique Drawing
First of all orthographic projections are drawn on one side of the sheet. The two main types of views (or "projections") used in drawings are: For convenience, the front view with circles was chosen as.
OBLIQUE PROJECTION DRAWING A simpler method of threedimensional
If you are a beginner in. Web types of views used in drawings. Web oblique projection is a type of parallel projection: All measurements drawn backwards are half the original measurement. Web oblique drawings are.
Oblique Projection Drawing The objects are not in perspective and so do not correspond to any view of an object that can be obtained in practice, but the technique yields somewhat convincing and useful. Web oblique drawing is a drawing in which the front view of the object is drawn to the correct size, and its side surfaces are drawn at an angle to give a pictorial appearance. Web in this video, i'll teach you all you need to know about oblique projection. Let's make an oblique drawing of a cube of size 4 x 4 x 4 on a piece of paper. If you are a beginner in. | 677.169 | 1 |
Hint: To solve the question, we have to apply the property of similar triangles to calculate the value of YZ.
Complete step-by-step Solution: \[\Delta ABC\sim \Delta XYZ\] symbolises that the triangles ABC and XYZ are similar triangles, which implies that the ratio of all the corresponding side of the given triangles is equal. AB, BC, CA of triangle ABC are corresponding sides of XY, YZ, ZX of triangle XYZ respectively. \[\Rightarrow \dfrac{AB}{XY}=\dfrac{BC}{YZ}=\dfrac{CA}{ZX}\] The given value of side BC of triangle ABC is equal to 5 cm. The given ratio of side AB of triangle ABC to side XY of triangle XYZ is equal to 2 : 3 By substituting the given values in the above expression, we get \[\dfrac{2}{3}=\dfrac{5}{YZ}=\dfrac{CA}{ZX}\] By solving the first part of the expression \[\dfrac{2}{3}=\dfrac{5}{YZ}\] we get, \[2YZ=5\times 3\] \[YZ=\dfrac{15}{2}=7.5\] cm. Thus, the value of YZ is equal to 7.5 cm.
Note: The possibility of mistake can be not applying the similar triangles property which is required to arrive at the solution. The other possibility of mistake can be misinterpreting the symbol of similarity to the symbol of congruence. The symbol for a similar triangle is one negation sign and the symbol for congruent triangle is two negation signs. The alternative method of solving the question can be by applying the direct formula for calculating \[YZ=\dfrac{XY}{AB}\times BC\]. Thus, the answer can be calculated quickly. | 677.169 | 1 |
Score High With 3 Important Trigonometry Fundamentals
Trigonometry is the study of the relationship between the ratios of the sides and angles of a right-angled triangle. Trigonometric ratios like sine, cosine, tangent, cotangent, secant, and cosecant are used to analyse this relationship. While the term trigonometry is a 16th-century Latin derivative, the idea of trigonometry was created by the Greek mathematician Hipparchus.
Basic Trigonometry
In the following information, we'll examine the theoretical underpinnings of trigonometry, its numerous identities, formulas,that follows, we'll examine the theoretical underpinnings of trigonometry, as well as its numerous identities and formulas and practical uses.
The sine, cosine, and tangent trigonometric operations are the three fundamental ones. These three fundamental functions provide the foundation for the cotangent, secant, and cosecant functions.
All concepts in trigonometry are based on these functions. Therefore, in order to completely understand trigonometry, we must first learn these functions.
Who will assist me with do my online class now that I'm at ease? Online education has several advantages for students and teachers since it allows for using students and teachers and for using it allows for the use of cutting-edge study methods that help pupils properly understand the subject material.
If the right-angled triangle's angle is, then
Perpendicular/hypotenuse = Sin
Base/hypotenuse = Cos
Tan = Base/Perpendicular.
The side that faces the right angle is called the hypotenuse.
The measurement of angles and problems involving angles are included in the fundamentals of trigonometry.
Sine, cosine, and tangent are the three fundamental operations of trigonometry.
These functions are used to build up all the crucial trigonometry concepts .Therefore, we first try to grab the knowledge about functions and their corresponding formulas to comprehend trigonometry.
Trigonometry identities are equations of trigonometric functions that hold in all circumstances. Along with solving trigonometry problems, trigonometry identities are widely utilised to understand basic mathematical ideas and solve other math problems.
Angle trigonometry
The trigonometric angles 0°, 30°, 45°, 60°, and 90° are frequently employed in trigonometry issues. Trigonometric ratios for certain angles, such as sine, cosine, and tangent, are easy to recall. We will also show the table that details each angle's values and proportions. To get these angles, we must create a right-angled triangle in which one of the acute angles is equal to the angle in trigonometry. The corresponding ratio will be discussed in relation to these angles.
Using the example of a triangle with a right angle
Sin θ = Perpendicular/Hypotenuse
or θ = sin-1 (P/H)
Similarly,
θ = cos-1 (Base/Hypotenuse)
θ = tan-1 (Perpendicular/Base)
Overview of the Delta Symbol
The letter delta is the fourth in the Greek alphabet. The basis for the delta was the Phoenician letter dalet. In mathematics, the delta is a fairly common symbol.
In mathematics, the delta symbol can be used to represent a number, function, set, or equation. Students can discover more about the delta symbol's mathematical significance here.
In mathematics, "change" or "the change" is generally denoted by an uppercase delta (). Imagine a situation where the variable x represents the motion of an object. As a result, "x" stands for "the change in movement." This mathematical interpretation of delta is used by researchers in a variety of scientific domains.
A lowercase delta () can be used to express an angle in any geometric shape in the language of geometry.
A lowercase delta () can be used to express an angle in any geometric shape in the language of geometry.
This is primarily due to the fact that geometry has its origins in Euclid's classical Greek texts. Greek letters were therefore used by mathematicians to denote their angle.
It is not necessary to comprehend or know the Greek alphabet. This is the case since the letters are only symbols for angles.
Even the tiniest alterations in one of a function's variables are taken into account in its derivative. The Roman letter "d" is sometimes used to represent a derivative.
Different from regular derivatives are partial derivatives. Even if there are numerous variations of the function, only one variable is considered in this situation. Without a doubt, the other elements continue to exist.
Capital Delta (Kronecker Delta)
Lowercase delta () has a much more specific use in advanced mathematics. Additionally, the lowercase delta in calculus denotes a change in a variable's value.
Think about the Kronecker delta scenario as an illustration. A connection between two integral variables can be seen in the Kronecker delta. This is 1 if the two variables are equal. Additionally, this equals 0 if the two variables are not same.
The application of trigonometry
In the past, trigonometry was employed in disciplines including surveying, astronomy, and construction. Its applications include
Physical sciences, astronomy, acoustics, navigation, electronics, oceanography, seismology, and a variety of other fields are included.
Long rivers' lengths can be estimated, a mountain's height can be determined, etc.
Spherical trigonometry has been used by astronomers to calculate the positions of the sun, moon, and stars.
One of the most important uses of trigonometry in daily life is calculating height and distance. The principles of trigonometry are frequently used in a number of fields, including the aviation department, navigation, criminology, marine biology, etc.
Mathematics is a methodical application of matter. It is assumed that a guy gets methodical or systematic as a result of the subject. Some of the qualities that mathematics nurtures include strong reasoning, originality, abstract or spatial thinking, critical thinking, problem-solving abilities, and even outstanding communication skills.
Numerous professions, including accounting, finance, banking, engineering, and software, as well as commonplace activities like driving, maintaining time, and cooking substantially rely on mathematics. These jobs require strong mathematical aptitude, and scientists working on scientific experiments need mathematical techniques. They act as a language to describe the efforts and successes of scientists.
For math class, you can receive college credit. You can develop analytical abilities in addition to obtaining a firm understanding of each mathematical topic.
Conclusion:
Mathematical symbols can be combined to represent concepts about the outside world. These symbols can occasionally signify numbers or be more abstract, signifying areas, symmetries, or groupings. Mathematical expressions are produced when these symbols are combined with mathematical operations like addition, subtraction, or multiplication, to name a few. | 677.169 | 1 |
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the dot operator " · " that is often used to designate this operation;[1] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
Definition
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The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.
In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate spaceRn{\displaystyle \mathbf {R} ^{n}}. In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.
Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique entry:
[13−5][4−2−1]=3.{\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}{\begin{bmatrix}4\\-2\\-1\end{bmatrix}}=3\,.}
Geometric definition
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Illustration showing how to find the angle between vectors using the dot product Calculating bond angles of a symmetrical tetrahedral molecular geometry using a dot product
In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction to which the arrow points. The magnitude of a vector a{\displaystyle \mathbf {a} } is denoted by ‖a‖{\displaystyle \left\|\mathbf {a} \right\|}. The dot product of two Euclidean vectors a{\displaystyle \mathbf {a} } and b{\displaystyle \mathbf {b} } is defined by[3][4][1]a⋅b=‖a‖‖b‖cosθ,{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,}
where θ{\displaystyle \theta } is the angle between a{\displaystyle \mathbf {a} } and b{\displaystyle \mathbf {b} }.
In terms of the geometric definition of the dot product, this can be rewritten as
ab=a⋅b^,{\displaystyle a_{b}=\mathbf {a} \cdot {\widehat {\mathbf {b} }},}
where b^=b/‖b‖{\displaystyle {\widehat {\mathbf {b} }}=\mathbf {b} /\left\|\mathbf {b} \right\|} is the unit vector in the direction of b{\displaystyle \mathbf {b} }.
These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that a⋅a{\displaystyle \mathbf {a} \cdot \mathbf {a} } is never negative, and is zero if and only if a=0{\displaystyle \mathbf {a} =\mathbf {0} }, the zero vector.
Equivalence of the definitions
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If e1,⋯,en{\displaystyle \mathbf {e} _{1},\cdots ,\mathbf {e} _{n}} are the standard basis vectors in Rn{\displaystyle \mathbf {R} ^{n}}, then we may write
a=[a1,…,an]=∑iaieib=[b1,…,bn]=∑ibiei.{\displaystyle {\begin{aligned}\mathbf {a} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {b} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}}
The vectors ei{\displaystyle \mathbf {e} _{i}} are an orthonormal basis, which means that they have unit length and are at right angles to each other. Since these vectors have unit length,
ei⋅ei=1{\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{i}=1}
and since they form right angles with each other, if i≠j{\displaystyle i\neq j},
ei⋅ej=0.{\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0.}
Thus in general, we can say that:
ei⋅ej=δij,{\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij},}
where δij{\displaystyle \delta _{ij}} is the Kronecker delta.
Vector components in an orthonormal basis
Also, by the geometric definition, for any vector ei{\displaystyle \mathbf {e} _{i}} and a vector a{\displaystyle \mathbf {a} }, we note that
a⋅ei=‖a‖‖ei‖cosθi=‖a‖cosθi=ai,{\displaystyle \mathbf {a} \cdot \mathbf {e} _{i}=\left\|\mathbf {a} \right\|\,\left\|\mathbf {e} _{i}\right\|\cos \theta _{i}=\left\|\mathbf {a} \right\|\cos \theta _{i}=a_{i},}
where ai{\displaystyle a_{i}} is the component of vector a{\displaystyle \mathbf {a} } in the direction of ei{\displaystyle \mathbf {e} _{i}}. The last step in the equality can be seen from the figure.
Triple product
The scalar triple product of three vectors is defined as
a⋅(b×c)=b⋅(c×a)=c⋅(a×b).{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ).}
Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
Generalizations
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Complex vectors
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For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself could be zero without the vector being the zero vector (e.g. this would happen with the vector a=[1i]{\displaystyle \mathbf {a} =[1\ i]}). This in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the dot product, through the alternative definition[12][2]a⋅b=∑iaibi¯,{\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i}{{a_{i}}\,{\overline {b_{i}}}},}
where bi¯{\displaystyle {\overline {b_{i}}}} is the complex conjugate of bi{\displaystyle b_{i}}. When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H:
In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in a{\displaystyle \mathbf {a} }. The dot product is not symmetric, since
a⋅b=b⋅a¯.{\displaystyle \mathbf {a} \cdot \mathbf {b} ={\overline {\mathbf {b} \cdot \mathbf {a} }}.}
The angle between two complex vectors is then given by
cosθ=Re(a⋅b)‖a‖‖b‖.{\displaystyle \cos \theta ={\frac {\operatorname {Re} (\mathbf {a} \cdot \mathbf {b} )}{\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}}.}
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
Functions
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The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length-n{\displaystyle n} vector u{\displaystyle u} is, then, a function with domain{k∈N:1≤k≤n}{\displaystyle \{k\in \mathbb {N} :1\leq k\leq n\}}, and ui{\displaystyle u_{i}} is a notation for the image of i{\displaystyle i} by the function/vector u{\displaystyle u}.
This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval[a, b]:[2]
Weight function
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Inner products can have a weight function (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions u(x){\displaystyle u(x)} and v(x){\displaystyle v(x)} with respect to the weight function r(x)>0{\displaystyle r(x)>0} is
Dyadics and matrices
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A double-dot product for matrices is the Frobenius inner product, which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices A{\displaystyle \mathbf {A} } and B{\displaystyle \mathbf {B} } of the same size: | 677.169 | 1 |
Exploring the Fascinating Conic Sections: An Introduction
AN INTRODUCTION TO CONIC SECTIONS
There exists a certain group of curves called Conic Sections that are conceptually kin in several astonishing ways. Each member of this group has a certain shape, and can be classified appropriately: as either a circle, an ellipse, a parabola, or a hyperbola. The term "Conic Section" can be applied to any one of these curves, and the study of one curve is not essential to the study of another. However, their correlation to each other is one of the more intriguing coincidences of mathematics.
There are four main types of conic sections: parabola, hyperbola, circle, and ellipse. The circle is sometimes categorized as a type of ellipse.
In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.
Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.
TYPES OF CONIC SECTIONS:
Parabola
Circle and ellipse
Hyperbola
PARABOLA
In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix). The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as
The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are similar.
Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola travelling parallel to its axis of symmetry is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected ("collimated") into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.
Summary
A parabola, showing arbitrary line (L), focus (F), and vertex (V).
CIRCLE AND ELLIPSE
CIRCLE
A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant. A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
ELLIPSE
In mathematics, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas.
Ellipses arise from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.
HYPERBOLAS
In mathematics a hyperbola is a type of smooth curve, four kinds of conic section, formed by the intersection of a plane and a cone. The other conic sections are the parabola, the ellipse, and the circle (the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a straight line on the surface of the cone makes with the axis of the cone. If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the plane intersects both halves of the double cone and the conic is a hyperbola.
Hyperbolas arise in practice in many ways: as the curve representing the function in the Cartesian plane, as the appearance of a circle viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet (one travelling too fast to ever return to the solar system), as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same), and so on. Each branch of the hyperbola consists of two arms which become straighter (lower curvature) further out from the center of the hyperbola.
Diagonally opposite arms one from each branch tend in the limit to a common line, called the asymptote of those two arms. There are therefore used in both relativity and quantum mechanics which is not Euclidean).
APPLICATIONS OF CONIC SECTIONS
The paraboloid shape of Archeocyathids produces conic sections on rock faces. Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem.
In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola. For certain fossils in paleontology, understanding conic sections can help understand the three-dimensional shape of certain organisms.
Students looking for free, top-notch essay and term paper samples on various topics. Additional materials, such as the best quotations, synonyms and word definitions to make your writing easier are also offered here. | 677.169 | 1 |
are vectors?
Vectors are mathematical objects that have both magnitude and direction. They are often represented as arrows in space, with the l...
Vectors are mathematical objects that have both magnitude and direction. They are often represented as arrows in space, with the length of the arrow representing the magnitude and the direction indicating the direction. Vectors are used in various fields such as physics, engineering, and computer science to represent quantities like velocity, force, and displacement. They can be added, subtracted, and multiplied by scalars to perform various operations.
What are collinear vectors?
Collinear vectors are vectors that lie on the same straight line or are parallel to each other. This means that they have the same...
Collinear vectors are vectors that lie on the same straight line or are parallel to each other. This means that they have the same direction or are in the opposite direction of each other. Collinear vectors can be scaled versions of each other, meaning one vector is a multiple of the other. In other words, collinear vectors have the same or opposite direction and are located on the same line or parallel lines.
How are vectors determined?
Vectors are determined by both magnitude and direction. The magnitude of a vector represents the length or size of the vector, whi...
Vectors are determined by both magnitude and direction. The magnitude of a vector represents the length or size of the vector, while the direction indicates the orientation of the vector in space. Vectors can be represented graphically as arrows, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction. Mathematically, vectors can be described using coordinates or components in a specific coordinate system.
What are lattice vectors?
Lattice vectors are a set of vectors that define the periodic structure of a crystal lattice. They represent the translation symme...
Lattice vectors are a set of vectors that define the periodic structure of a crystal lattice. They represent the translation symmetry of the lattice and can be used to generate all the points in the lattice by adding integer multiples of the lattice vectors to a reference point. In a 3D crystal lattice, there are typically three lattice vectors that are linearly independent and form the basis for the lattice. The lattice vectors are essential for describing the crystal structure and understanding the physical properties of materials.
Are the vectors collinear?
To determine if the vectors are collinear, we need to check if one vector is a scalar multiple of the other. If the vectors are co...
To determine if the vectors are collinear, we need to check if one vector is a scalar multiple of the other. If the vectors are collinear, then one vector can be obtained by multiplying the other vector by a scalar. If the vectors are not collinear, then they will not be scalar multiples of each other.
Source:AI generated from FAQ.net
What are basis vectors?
Basis vectors are a set of linearly independent vectors that can be used to represent any vector in a given vector space through l...
Basis vectors are a set of linearly independent vectors that can be used to represent any vector in a given vector space through linear combinations. They form the building blocks for expressing any vector in the space. In a 2D space, the basis vectors are typically denoted as i and j, while in a 3D space, they are denoted as i, j, and k. Basis vectors are essential for understanding and working with vector spaces in linear algebra and are fundamental to many mathematical and physical concepts.
Source:AI generated from FAQ.net
What are parallel vectors?
Parallel vectors are vectors that have the same or opposite direction, but may have different magnitudes. In other words, if two v...
Parallel vectors are vectors that have the same or opposite direction, but may have different magnitudes. In other words, if two vectors are parallel, they either point in the same direction or in exactly opposite directions. This means that one vector is a scalar multiple of the other. For example, if vector A is parallel to vector B, then vector A = k * vector B, where k is a scalar.
Source:AI generated from FAQ.net
How do vectors intersect?
Vectors intersect when they share a common point in space. This point is known as the point of intersection. To determine if two v...
Vectors intersect when they share a common point in space. This point is known as the point of intersection. To determine if two vectors intersect, we can set their parametric equations equal to each other and solve for the variables. If the resulting values satisfy both equations, then the vectors intersect at that point. If the vectors are parallel or skew (non-intersecting and non-parallel), they do not intersect.
What are the center vectors?
Center vectors are the vectors that represent the average of a set of vectors in a given space. They can be calculated by adding a...
Center vectors are the vectors that represent the average of a set of vectors in a given space. They can be calculated by adding all the vectors together and dividing by the total number of vectors. Center vectors are useful in machine learning for tasks such as clustering, where they can be used to represent the center or average point of a cluster of data points.
Are two identical vectors collinear?
Yes, two identical vectors are collinear. Collinear vectors are vectors that lie on the same line or are parallel to each other. S...
Yes, two identical vectors are collinear. Collinear vectors are vectors that lie on the same line or are parallel to each other. Since identical vectors have the same direction and magnitude, they are considered collinear.
What are linearly independent vectors?
Linearly independent vectors are a set of vectors where none of the vectors can be written as a linear combination of the others....
Linearly independent vectors are a set of vectors where none of the vectors can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a scalar multiple of another vector in the set. If a set of vectors is linearly independent, then the coefficients of the linear combination that equals zero must all be zero. This property is important in linear algebra as it allows for unique solutions to systems of linear equations.
What are vectors in mathematics?
In mathematics, vectors are quantities that have both magnitude and direction. They are often represented as arrows in a coordinat...
In mathematics, vectors are quantities that have both magnitude and direction. They are often represented as arrows in a coordinate system, with the length of the arrow representing the magnitude and the direction indicating the direction. Vectors are used to describe physical quantities such as force, velocity, and acceleration, as well as in various mathematical operations such as addition, subtraction, and scalar multiplication. Vectors play a crucial role in many branches of mathematics, including linear algebra, calculus, and | 677.169 | 1 |
Polar Coordinates Assignment Help
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If you are studying mathematics, you will soon learn about polar coordinates. This is the most fundamental lessons on higher geometry where you will learn to indicate the position of a point or geometric figure in the 2-dimensional plane. There are lots of equations for different geometric figures like circle, ellipse, parabola, and hyperbola which you have to know in-depth. There will be assignments too on that topic and they will not be easier ones to complete. Your expert knowledge will be required to solve highly complicated topic in polar coordinates. But never worry, if you find any topic really tough or if for any other reasons, it seems that completing the assignment within the deadline will be tough, contact BookMyEssay for professional Polar Coordinates assignment help.
In polar coordinates assignments, three fields of mathematics are important: algebra, geometry, and trigonometry. Each point in a polar coordinate can be described with radial coordinate or r and angular coordinate or θ (theta). The latter one is also known as Azimuth or polar angle. You may require changing Polar coordinates to Cartesian coordinates and vice versa. This follows a particular rule. The related equations of shifting from one system to another are as follows:
You know how to use these equations to find any unknown quantities depicted here, i.e. x, y, and θ.
Polar coordinate geometry is a part of applied mathematics which is often used in depicting different positions in space like a position of a planet or star. Navigators, meteorologists, and military personnel also use polar coordinates systems and formulas for understanding the positions of different objects in distant places. These days, computer applications play a vital part in polar coordinates. So, you may require understanding some computer applications in this realm.
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Polar Coordinates: Some Important Aspects
The coordinates system that expresses a point in a two-dimensional space in terms of radius and angle with respect to the origin are known as polar coordinates. The angle always rotates or expressed with respect to the positive horizontal axis which is known as polar axis.
In the above figure: the point P is located at (r, θ) where r is the radius of the point from the origin or distance of the point from the origin and θ is the angle of the point with respect to the X-axis and it is always in the right direction (positive direction). In advanced and applied mathematics, the polar angle is mostly expressed in radians. See the image below to understand it better:
Starting from 0, it ends at 2π where π = 180 degrees. Here, π/2 = 90 degrees. Apart from general formulas and problems, you have to deal with many other topics:
Lines in two dimensions, Properties of parallel lines, Properties of perpendicular lines, Properties of different geometrical figures, and practical applications of this form polar coordinates.
Polar Coordinates Assignment help
Our Polar Coordinate assignment help can be approached any time. Our writers take care of all important aspects of an assignment writing Tip's, i.e. the deadline, the guidelines, and quality. As such, you will always get your task completed before the deadline. They follow the guidelines you provide closely and never ignore any aspect of it. At last, they ensure that:
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Is arccos 0 an angle?
The arccos function is the inverse of the cosine function. It returns the angle whose cosine is a given number….For y = arccos x :
Range
0 ≤ y ≤ π 0 ° ≤ y ≤ 180 °
Domain
− 1 ≤ x ≤ 1
What is arccos 0.5 solution?
arc cosine(0.5) = 60°
What is the exact value of arccos 1?
The exact value of cos-1(1) is 0 .
What does arccos equal?
Arccosine is the inverse of the cosine function and thus it is one of the inverse trigonometric functions. Arccosine is pronounced as "arc cosine". Arccosine of x can also be written as "acosx" (or) "cos-1x" or "arccos". If f and f-1 are inverse functions of each other, then f(x) = y ⇒ x = f-1(y).
How do you convert arccos to degrees?
Arccos(x) calculator. Inverse cosine calculator….Arccos table.
x
arccos(x)
degrees
radians
-0.5
120°
2π/3
0
90°
π/2
0.5
60°
π/3
What is arccos on a calculator?
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, the arccos is the inverse of the cosine. It is normally represented by arccos(θ) or cos-1(θ).
What is the arccos of 2?
Since the cosine function has output values from -1 to 1, the arccosine function has input values from -1 to 1. So arccos x is undefined for x=2.
What is arccos calculator?
Arccos function The arccos is used to obtain an angle from the cosine trigonometric ratio, which is the ratio between the side adjacent to the angle and the hypotenuse in a right triangle. The function spans from -1 to 1, and so do the results from our arccos calculator.
Is arccot the same as cot 1?
It turns out that arctan and cot are really separate things: cot(x) = 1/tan(x) , so cotangent is basically the reciprocal of a tangent, or, in other words, the multiplicative inverse. arctan(x) is the angle whose tangent is x.
What is the arccos (x)?
The Arccos (x) is also referred to as the inverse cosine calculator. It is used to calculate the inverse of a cosine. This calculator has a single text field and there controls.
What is the value of arccos = cos-1?
arccos (x) = cos -1 (x) For example, If the cosine of 60° is 0.5: cos (60°) = 0.5 Then the arccos of 0.5 is 60°:
What is the arccos smart caddie?
ARCCOS CADDIE. Smart Grips. Featuring the world's smartest caddie, powered by Artificial Intelligence, Arccos Smart Grips allow golfers to automatically record and analyze every shot they take on the course with no tagging, tapping or other disruptions.
How does arccos work?
Arccos seamlessly captures thousands of data points during every round you play and provides unequaled insights about how far to hit each shot, which club to use in every situation, and what skills to practice, helping golfers of all skill levels improve faster than ever. | 677.169 | 1 |
Prove the HL Theorem without angle measures
or Pythagorean theorem. cannot
include HL theorem in proof
Homework Answers
Answer #1
HL Theorem: If the hypotenuse and one leg of a right triangle
are equal to the hypotenuse and one leg of another right triangle,
then the two right triangles are congruent.
Triangle ABC Triangle DEF
Area of triangle ABC = AB x BC (For right-angle triangle area =
base x height)
Area of triangle DEF = DE x EF
If they are congruent the area is also the same.
AB x BC = DE x EF (One leg of the triangle is the same AB = DE, and
the hypotenuse is also same )
BC = EF
Now, we have all three sides are the same of above two triangles.
Now we can say they are congruent to each other. | 677.169 | 1 |
Circles as Regular Polygons
As we said before, the circle is special because we can approach it as a regular polygon with 36 (or more) vertices. We will use exactly 36 vertices for this example.
So, if we want the robot to do a circle (clockwise) we need to give the following commands (note: the exterior angle values may be difficult to see in the image but each is 10 degrees):
Keep in mind that the S mustn't be a big number because otherwise your circle will be very large! Choose small numbers for the S values. For example, in the next example S = 8 ticks.
The following SimpleIDE code programs the ActivityBot to make a circle with radius of approximately 15 cm:
Start a new SimpleIDE Project
Enter in the code below and make sure load it to your robot with the "Load EEPROM & Run" button
The question that might arises from this example is: what length should the S be in order to make a circle with a specific radius (R)? In other words, can I calculate the S as a function of R? Math theory shows us that "Cosines Law" is the answer to this. Looking at the triangle OA1A2 (below) we can see that:
So now we are almost ready for the final algorithm that will be able to create any circle with a given radius R, or create a polygon within a circle of radius R. | 677.169 | 1 |
Children should see that sometimes it is easier to scale numbers in one 'direction' than the other, and that they can choose the most efficient method. 20) given 2 squares and 2 triangles, find the area of the red triangle. Web geometry basics will teach you the 5 simple rules needed to answer basic geometry questions, as well as give you the foundations to build as you work through the different geometry topics. Web designed to help children to visualise the relationships between numbers when answering word problems involving ratio and scaling quantities. These problems involve proving that two or more shapes are similar.
20) given 2 squares and 2 triangles, find the area of the red triangle. Web calculating the volume of cubes, prisms, and other 3d shapes. (the 12 dots are equally spaced; (this one feels like an instant classic to me.)
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World's Hardest Easy Geometry Problem Step by Step Solution
For more resources involving word problems click here. Contains over 160 complex problems with answers, hints, and detailed solutions. There is also an example of a geometry word problem that uses similar triangles. Having basic.
How to Write a Congruent Triangles Geometry Proof 7 Steps
Find the value of r. In these lessons, we will learn to solve geometry math problems that involve perimeter. These problems involve proving that two or more shapes are similar. Understanding and using the pythagorean.
Geometry Rotation Worksheet
Web challenging problems in geometry is organized into three main parts: 22) find the area of the orange triangle. 21) find the area of the red region. You can see how to solve geometry word.
Shapes Worksheet Word Problems Grade 3 Math Math word problems, Word
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Web 16) find the area of the grey rectangle. Measuring, classifying, constructing and calculating angles. (this one feels like an instant classic to me.) 18) find the area of the blue rectangle. An isosceles triangle.
The only point used inside the circle is the centre.) "unfortunately, my favourite one of the six is the only one i didn't come up with myself," says catriona, "the dark blue one." 2. Web advanced areas of triangles (1/2 absinc) past exam booklets with fully solutions for transformations, area, perimeter, circle theorems, pythagoras, trig & more. We feel that announcing the technique to be used stifles creativity and destroys a good part ofthe fun ofproblem solving. Web below you will find practice worksheets for skills including using formulas, working with 2d shapes, working with 3d shapes, the coordinate plane, finding volume and surface area, lines and angles, transformations, the pythagorean theorem, word. Web algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.classically, it studies zeros of multivariate polynomials;
Contains over 160 complex problems with answers, hints, and detailed solutions. The modern approach generalizes this in a few different aspects. These problems involve finding the area and perimeter of.
Having Basic Algebra Knowledge Is Required To Solve Geometry Problems.
These problems involve proving that two or more shapes are congruent. These problems involve finding the area and perimeter of. Geometry math problems involving area. Children should see that sometimes it is easier to scale numbers in one 'direction' than the other, and that they can choose the most efficient method.
Web Geometry Problems Often Involve Shapes, Sizes, Positions, And The Properties Of Space.
Web some of the common types of geometry problems include: (this one feels like an instant classic to me.) Web geometry solver that solves problems step by step. Proofs and postulates worksheet practice exercises (w/ solutions) topics include triangle characteristics, quadrilaterals, circles, midpoints, sas, and more.
What Fraction Of Each Circle Is Shaded?
18) find the area of the blue rectangle. 21) find the area of the red region. Web designed to help children to visualise the relationships between numbers when answering word problems involving ratio and scaling quantities. Classifying types of triangles, quadrilaterals and polygons.
The Only Point Used Inside The Circle Is The Centre.) "Unfortunately, My Favourite One Of The Six Is The Only One I Didn't Come Up With Myself," Says Catriona, "The Dark Blue One." 2.
The modern approach generalizes this in a few different aspects. 22) find the area of the orange triangle. 19) red line is tangent to blue circle. An isosceles triangle has an altitude equal to 12 inches and a perimeter equal to 36 inches.
Sometimes it involves common solids like spheres, cylinders, rectangular parallelepiped, and cubes. You can see how to solve geometry word problems in the following examples: Web advanced areas of triangles (1/2 absinc) past exam booklets with fully solutions for transformations, area, perimeter, circle theorems, pythagoras, trig & more. The only point used inside the circle is the centre.) "unfortunately, my favourite one of the six is the only one i didn't come up with myself," says catriona, "the dark blue one." 2. Children should see that sometimes it is easier to scale numbers in one 'direction' than the other, and that they can choose the most efficient method. | 677.169 | 1 |
Maximum area of a triangle inscribed in another triangle?
In summary, to maximize the area of BDE, the length of DE should be (a/2). To maximize the perimeter of BDE, the length of DE should be (a/2) + (a-a/x).
Jan 31, 2015
#1
Daveigh
5
0
Homework Statement
[/B]
Hello!
I have this question which I don't quite know how to solve...
ABC is an equilateral triangle - the length of its sides equal to (a).
DE is parallel to BC
1. What length should DE be to achieve the largest possible area of triangle BDE?
2. What length should DE be to achieve the smallest possible perimeter of triangle BDE?
How should this be done (step-by-step, my knowledge of math is quite basic)?
The attempt at a solutionAlso, I didn't manage to solve the second question about the perimeter.
If anyone could explain and show the correct answer for comparison, I would be very grateful.The method looks fine and the answer is correct, that looks good.
It is easier if you use the formula A=1/2 (DE)*(height) because then you avoid dealing with odd angles.
Daveigh said:
Also, I didn't manage to solve the second question about the perimeter.
What did you get as perimeter?
Jan 31, 2015
#3
Daveigh
5
0You can factor out a from the expression of the perimeter. Note that a/x + (a-a/x) = a.
Expand the parentheses under the square root and simplify before taking the derivative.
There are two other tricks that can help:
The minimum of f(x) is also a minimum of f(x)+b with some constant b.
The minimum of f(x) is also a minimum of f2(x). The minimum of f2(x), if not zero, is a minimum OR maximum of f(x) depending on the sign of f(x).
But there is an even better approach: ehild's hint gives you the sum of two sides of the triangle, and it is constant. You just have to minimize the length of a specific side (which one?), and this can be done without calculations.
Related to Maximum area of a triangle inscribed in another triangle?
1. What is the maximum area of a triangle inscribed in another triangle?
The maximum area of a triangle inscribed in another triangle is half of the area of the larger triangle. This is known as the "half base times height" formula.
2. How do you find the maximum area of a triangle inscribed in another triangle?
To find the maximum area, you need to first find the length of the base and the height of the inscribed triangle. Then, you can use the formula "1/2 * base * height" to calculate the maximum area.
3. Can the maximum area of a triangle inscribed in another triangle be greater than half of the area of the larger triangle?
No, the maximum area of a triangle inscribed in another triangle can never be greater than half of the area of the larger triangle. This is because the inscribed triangle is always a fraction of the larger triangle.
4. What is the significance of finding the maximum area of a triangle inscribed in another triangle?
Finding the maximum area of a triangle inscribed in another triangle is useful in various applications, such as optimizing the use of space in a given area. It can also help in solving geometric problems and proving certain theorems.
5. Are there any other ways to find the maximum area of a triangle inscribed in another triangle?
Yes, there are other methods such as using the Pythagorean theorem or trigonometric ratios to find the length of the base and height of the inscribed triangle. However, the "half base times height" formula is the most commonly used method. | 677.169 | 1 |
The ratio is calculated as 6.1/6.7≈0.916.1/6.7 \approx 0.916.1/6.7≈0.91 based on the asksia-ll calculation list
step 3
Compare the calculated ratio to the ratios given in the table for 55∘55^{\circ}55∘, 65∘65^{\circ}65∘, and 75∘75^{\circ}75∘
step 4
The ratio closest to 0.910.910.91 in the table is for the angle 65∘65^{\circ}65∘, which has a ratio of 0.910.910.91
step 5
Therefore, the measure of angle D is approximately 65∘65^{\circ}65∘
[question number] Answer
B
Key Concept
Using ratios to determine angle measures in right triangles
Explanation
The ratio of the lengths of sides in a right triangle is related to the measure of its angles. By comparing the calculated ratio of side lengths to known ratios for specific angles, we can approximate the measure of an angle. | 677.169 | 1 |
Abstract Geometry
Evolving Technical Notes
Elementary Shapes
We'll be working with several types of objects: points (P), line segments (L), equilateral triangles (T), squares (Sq), and circles (Cir). A line segment that connects points a and b is denoted as [a, b] and is treated as a singular object. The notation (a,b) will specifically refer to a pair of objects. Our primary relation of interest is connectivity, which we will represent with the letter C.
For instance, a triangle constructed from three interconnected points a, b, and c can be described as:
<a, b, c; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a) >
The triangle's reflectional and rotational symmetries align with the Abstract Geometry morphisms of the structure into itself. Such a morphism of an object into itself is termed an automorphism.
Drawing from Tarski's axiomatization of geometry rooted in predicate logic, we'll introduce the special relation of 'betweenness,' denoted as B(a,b,c). This implies that point 'b' lies between points 'a' and 'c.' To define all points lying between 'a' and 'b,' set notations and pattern variables are used. A set comprising objects a,b,c,d is described as {a,b,c,d}. By using pattern variables like {X} followed by one or more predicates containing variable X, we can characterize a set of all 'X' meeting the predicate conditions. This allows us to define a line segment [a,b] as the set of all points 'X' situated between 'a' and 'b':
[a, b] = <{X}, B(a, X, b)>
Here, the set {X} functions as a distinct object. To account for situations where one set is contained within another or one shape is contained within another, we'll centered regular polygons, we'll regard an object as comprising one object "a" and set {X}. Such an object is portrayed as (a, {X}) or a, {X}. Using this representation, we can discuss a pattern shared by both circles and regular polygons:
< (a, {X}); R(a, X) >
< a, {X}; R(a, X) >
Depending on the cardinality of set {X}, this pattern could represent either a polygon or a circle. This pattern is what we'll term an "abstract centroid" with the abstract radius R.
For an object (a, {X}) to be identified as a circle, we'll need to append another predicate:
< (a, {X}); R(a, X), Cir(a, {X}) >
We have the flexibility to omit the relation R(a, X) and/or introduce details about the circle's radius, which will be elaborated on later. Comparable notations can be employed for triangles, squares, and regular polygons that designate object a as their central point:
< (a, {X}); Sq(a, {X}) >
< (a, {b, c, d, e}); Sq(a, {b, c, d, e}) >
Using these notations, a square inscribed in a circle can be described as:
< a, {X}, {Y}; Sq(a, {X}), Cir(a, {Y}), in({X, Y}) >
Part-Whole Relation and Connectivity
A line segment or a triangle are objects in their own right. We will describe this metamorphosis as follows:
This leads to a reflection on the versatile nature of the connectivity relation 'C'. It's worth exploring the multiple interpretations of this relation, such as two points linked by a line or two segments intersecting at a point. For instance, connecting four triangles side by side can give rise to a tetrahedron, while linking the edges of eight triangles can form an octahedron. Ambiguity is indeed an intrinsic characteristic of human language.
Lengths, Angles, Ratios, Areas, and Perimeters
In symbolic logic, numeric properties of objects would normally be described in terms of functions. Say the fact that the angle between segments a and b is equal to A would be described as Ang(a,b) = A.
Oftentimes we care about the equality of two angles or ratios on different shapes rather than about the specific values. Moreover, equality should not be precise. If our angles and ratios are predicates we can still think of morphism as mapping that preserves predicates. For that reason, we use non-standard notations and describe angles, lengths, and ratios as labeled relations. The fact that the angle between segments a and b is equal to A would be described as AngA(a, b).
We will prefix the relation for length with Len, for ratio with Rt, Ar for area and Per for perimeter. We endow a number with the type of measurement which this number describes. The purpose of this deviation from standard notations of logic is to keep the picture of morphism and automorphism simple. Using such notations we can describe circle of radius 1 and perimeter 2???? as follows:
< (a, {X}); Cir(a, {X}), Len1(a, X), Per2????((a, {X}))>
Here is the advantage these esoteric notations bring to our framework. Consider a circle c and a square sq with the same area D. We represent area measurement as a predicate ArD. Because the circle and the square have a common predicate ArD they have one partial isomorphism
<c; Cir(c), ArD(c) >
|
<sq; Sq(sq), ArD(sq) >
This morphism preserves patten
<X; ArD(X) >
We could have described such morphism with standard logic notations but that would look complicated.
Sacred Geometry and Abstract Geometry
From the standpoint of modern Math, squaring of a circle is just another geometrical problem. From the standpoint of Sacred Geometry, it is an important mystery. In Sacred Geometry shapes have meaning. A square may symbolize material form or rational intellect traditionally or left brain intellect. A circle may symbolize soul, spirit or right brain intellect. When such is the meaning of the symbols then squaring a circle may mean harmonizing matter with soul or rational male intellect with female right brain intellect.
Why are squares and circles with the same area considered to be harmonized? Abstract Geometry is the only framework that can provide a meaningful mathematical answer to this question. Abstract Geometry elevates the relation between square and circle of the same area to the relation between two squares (or between two circles) because all of these relations are Abstract Geometry partial morphisms. Abstract Geometry describes Harmony implied in Sacred Geometry with the aid of partial morphisms.
The total Harmony associated with the system is a set of partial morphisms along with their degrees.
As morphisms have degrees assigned to them, one morphism can create more Harmony than the other. A square and a circle with the same center are more harmonious then ones which have different centers. Consider moving square and circle from previous subsection to a common center a and making a new partial morphism:
<c, a; Cir(c), Center(a, c), ArD(c) >
| |
<sq, a; Sq(sq), Center(a, sq), ArD(sq) >
This morphism preserves pattern:
<X, Y; Center(Y, X), ArD(X) >
This morphism has degree 4 as it maps two objects across 2 predicates. The original morphism had degree 1. So, the latter shape contains more Harmony than the former.
Abstract Geometry reveals important mysteries of Platonic Solids related to their duality. If we use the same predicate C for different types of connected objects we may discover some fascinating morphism between Platonic Solids and their duals. E.g. an octahedron is dual to a cube. An octahedron described as 8 triangles connected side by side will have a morphism to a cube described as 8 points connected by lines.
Why do morphisms discovered with the help of Abstract Geometry have some mystical meaning? In the next post on beautyandai.com I'll argue that Abstract Geometry captures the essence of the primordial forms discussed by Plato. He rightfully compared Forms to shapes drawn by geometers because the second are images of the first.
By replacing the relation R with Len1, this structure can be transformed into a circle of radius 1 with an inscribed triangle.
It's vital to emphasize that Abstract Geometry comprises structures tailored to describe images, specifically focusing on their geometric properties. Similar to structures that depict situations and imagery, Abstract Geometry's structures should remain adaptable. Predicates can be added, removed, or altered, and one structure can be converted to another using transformation rules. If Abstract Geometry has any theorems, they would manifest as transformation rules. | 677.169 | 1 |
Shape Octagon
Shape Octagon - Web simple or complex a simple polygon has only one boundary, and it doesn't cross over itself. However, octagons can vary widely in their shapes. It is classified as a polygon, as well as an octagon. Thus, an octagon has eight edges, eight angles and eight vertices. Web the octagon shape will be introduced to your student around the thrid grade. Web an octagon refers to a polygon in geometry having 8 sides and 8 angles. Web definitions octagon is a polygon with eight sides and eight vertices. Web whether your want a traditional rectangular picnic table with attached benches, or an octagonal picnic table, or separated. Web trace and color octagon shapes. An octagon, like any other polygon, can be.
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A polygon of eight angles and eight sides. Web an octagon is a closed 2d shape that has 8 sides and 8 angles. Now, what is a polygon? Web whether your want a traditional rectangular picnic table with attached benches, or an octagonal picnic table, or separated. Regular or irregular when all angles are equal and all.
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I belong to the polygon family. The style became popular in the united states and canada following the publication of orson. Web an octagon is a closed 2d shape that has 8 sides and 8 angles. Octagon shape the shape of an octagon depends on the type of octagon. Web properties in the above given octagon, ∠a + ∠b +.
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A polygon of eight angles and eight sides. It is a closed plane. I am an octagon shape. Web in this octagon area calculator, you will also find the answers to the following questions: Web the octagon shape will be introduced to your student around the thrid grade.
Mathematics how to draw a regular polygon octagon using 360 degrees
Web this is a list of octagon houses. A polygon of eight angles and eight sides. Web properties in the above given octagon, ∠a + ∠b + ∠c + ∠d + ∠e + ∠f + ∠g + ∠h = 1080°. An octagon, like any other polygon, can be. Now, what is a polygon?
Web definitions octagon is a polygon with eight sides and eight vertices. Web simple or complex a simple polygon has only one boundary, and it doesn't cross over itself. Thus, an octagon has eight edges, eight angles and eight vertices. Web an octagon, by definition, is a polygon that has eight sides. Web an octagon is a polygon containing eight sides. The style became popular in the united states and canada following the publication of orson. I belong to the polygon family. I am an octagon shape. Web this is a list of octagon houses. The main reason will be to ensure your child is. Web an octagon can be defined as a polygon with eight sides, eight interior angles, and eight vertices. Now, what is a polygon? Web the amish hancrafted amish poly octagon picnic table is carefully constructed in lancaster, pa to provide you a lifetime of. It is classified as a polygon, as well as an octagon. First, have a play with an octagon: Web an octagon is a closed shape with eight sides and eight angles. Web trace and color octagon shapes. It is a closed plane. Web properties in the above given octagon, ∠a + ∠b + ∠c + ∠d + ∠e + ∠f + ∠g + ∠h = 1080°. Thus, as it has got 8 sides and angles, the vertices and.
The Style Became Popular In The United States And Canada Following The Publication Of Orson.
Thus, an octagon has eight edges, eight angles and eight vertices. Now, what is a polygon? Web trace and color octagon shapes. Web an octagon can be defined as a polygon with eight sides, eight interior angles, and eight vertices.
Web An Octagon Is A Closed 2D Shape That Has 8 Sides And 8 Angles.
Web an octagon is a polygon containing eight sides. Web an octagon refers to a polygon in geometry having 8 sides and 8 angles. I am an octagon shape. Web definitions octagon is a polygon with eight sides and eight vertices.
Web in this octagon area calculator, you will also find the answers to the following questions: Built with a great shape. It is classified as a polygon, as well as an octagon. Web the octagon shape will be introduced to your student around the thrid grade.
However, Octagons Can Vary Widely In Their Shapes.
Web this is a list of octagon houses. Web an octagon is a closed shape with eight sides and eight angles. What is an octagon, how. I belong to the polygon family. | 677.169 | 1 |
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Text solutionVerified
Step 1. Given that we have two triangles ΔABC and ΔDEF, where AB=DF and ∠A=∠D.
Step 2. According to SAS axiom, we can prove that the two triangles are congruent if one of the sides and the included angles of one triangle is equal to the corresponding side and included angles of the other triangle.
Step 3. Therefore, we can write AC=DE as the included side and ∠A=∠D as the included angles are equal.
Step 4. Thus, the correct option is (b) AC=DE as per the given situations in ΔABC and ΔDEF.
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Michael Discussed | 677.169 | 1 |
Class 8 Courses
If the angle of intersection at a point the angle of intersection at a point where the two circles with radii $5 \mathrm{~cm}$ and $12 \mathrm{~cm}$ intersect is $90^{\circ}$, then the length (in $\mathrm{cm}$ ) of their common chord is : | 677.169 | 1 |
Suppose p and q are picked out of the number line, where p is lying to the right of q. Then, p > q All numbers on the number line to the left of a reference point are smaller than the number on the reference point. Similarly, all numbers on the number line to the right of a reference point are greater than the number on the reference point. | 677.169 | 1 |
Define an ear to be a corner of a polygon that sticks out. More rigorously, three adjacent vertices form an ear if the angle they form is convex and the triangle they form does not contain any of the polygon's other points. It can be shown that any polygon with more than three points has at least two ears.
The method used by this program is:
Do While (# vertices > 3)
Find an ear.
Add the ear to the triangle list.
Remove the ear's middle vertex from the polygon. | 677.169 | 1 |
1.5:This lesson works with:
Grade level:
Approx. time required:
Shapes withStep 2
Square code
To appreciate the value of a for loop, first think about the code for driving in a square. Since a square has four equal sides, you would need to go forward then turn left or right four times.
Use the driving blocks to write a code that drives Zumi in a square. You have done this before so this should be a review!
Step 3
Loops
You have probably noticed that you repeat the same section of code four times. That's a lot of blocks! To get around this, you can use a loop, which lets you repeat parts of your code. They're a great shortcut — without them, you would always have to use the same blocks over and over again. They are called loops because the last statement loops around back to the beginning. The loop blocks that help you repeat code are found in the "Loops" menu.
Step 4
Repeat block
Any code attached to a repeat block will repeat as many times as you set. This means you can choose how many times your code will repeat! Here is the same square code written with a repeat block.
Step 5
More shapes!
What other shapes can you make with loops? The square is the easiest because each turn is 90 degrees, but what about other shapes? How many sides does a triangle have? This particular triangle will be an equilateral triangle, meaning all three sides are of equal length and every angle is the same number of degrees. Here's a hint: the exterior angles of all regular polygons, or shapes with equal sides, always add up to 360 degrees.
There are many other shapes you can teach Zumi. Try pentagons, hexagons, or octagons! You may need a pencil and paper to figure out how many degrees you need to turn. Remember, the number of degrees multiplied by the number of sides should be equal to 360 degrees.
Step 6
More Shapes - Solutions
Once you have tried making the different shapes with Zumi, try comparing your code with the solutions below!
Pentagon:
Hexagon:
Octagon:
Step 7
Review
In this lesson, you learned how to use the repeat block! The repeat block is helpful when you want to repeat code a certain number of times. In the next lesson, you will learn about conditionals and how we can make loops run as long as something is true. | 677.169 | 1 |
Category: NCERT Class 9 Maths Notes
CBSE Class 9 Maths Chapter 15 Notes Probability Probability Class 9 Notes Understanding the Lesson 1. Experiment: A procedure which produces some well-defined possible outcomes. 2. Random experiment: An experiment which when performed produces one of the several possible outcomes called a random experiment. 3. Trial: When we perform an experiment it is called a… Read more
CBSE Class 9 Maths Chapter 12 Notes Constructions Constructions Class 9 Notes Understanding the Lesson Geometrical construction means using only a ruler and a pair of compasses as geometrical instruments. Protractor may be used for drawing non-standard angles. Construction of bisector of a line segment using compass Draw a line segment, say AB. With both… Read more
CBSE Class 9 Maths Chapter 11 Notes Circles Circles Class 9 Notes Understanding the Lesson Circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane. The fixed point is called the centre O and the given distance is called the radius r of the circle.… Read more
CBSE Class 9 Maths Chapter 9 Notes Quadrilaterals Quadrilaterals Class 9 Notes Understanding the Lesson Quadrilateral A plane figure bounded by four line segments is called quadrilateral. Properties: It has four sides. It has four vertices or comers. It has two diagonals. The sum of four interior angles is equal to 360°. In quadrilateral ABCD,… Read more
CBSE Class 9 Maths Chapter 8 Notes Linear Equations in Two Variables Linear Equations in Two Variables Class 9 Notes Understanding the Lesson 1. Equation: An equation is a mathematical statement that two things are equal. It consists of two expressions one on each side of an equals sign. For example,7x + 9 = 0… Read more
CBSE Class 9 Maths Chapter 6 Notes Coordinate Geometry Coordinate Geometry Class 9 Notes Understanding the Lesson Rene Descartes was a French mathematician. He introduced an idea of Carterian Coordinate System for describing the position of a point in a plane. The idea which has given rise to an important branch of Mathematics known as… Read more | 677.169 | 1 |
Difference Between Protractor and Compass
Table of Contents
Key Differences
A protractor is primarily used for measuring angles, commonly found in educational settings where geometry is taught. On the other hand, a compass is used for drawing circles or arcs, crucial in both educational and professional drafting environments.
Protractors typically feature a semicircular design marked with degrees from 0 to 180. Whereas, compasses consist of two legs connected at a pivot; one leg has a point to anchor the tool, and the other holds a pencil to draw.
The material of a protractor is usually transparent plastic or glass to easily see the base lines and angles being measured. Conversely, compasses are often made of metal for stability and precision, with some high-end models also incorporating adjustment mechanisms for more accurate control.
While protractors provide static measurements of angles, compasses offer dynamic use, allowing the user to adjust the radius and create multiple sizes of circles or arcs based on the distance between the legs.
Protractors are often used in conjunction with rulers and straightedges to measure and draw precise angles. In contrast, compasses can be used independently to construct the needed geometric figures without additional tools.
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Comparison Chart
Primary Use
Measuring angles
Drawing circles or arcs
Design
Semicircular, marked in degrees
Two legs with a pivot, one leg with a pencil
Material
Transparent plastic or glass
Typically metal
Method of Use
Static measurement
Dynamic use with adjustable radius
Dependency on Tools
Used with rulers and straightedges
Can be used independently
Compare with Definitions
Protractor
Collaborates with other tools for accurate measurements.
She aligned her protractor with the straightedge to measure the angle precisely.
Compass
Often used in drafting and design.
The architect used a compass to sketch the initial design of the roundel.
Protractor
Usually semicircular in shape and marked in degrees.
Her protractor was made of clear plastic, making it easy to read the degrees.
Compass
A tool used for drawing circles or arcs.
He adjusted the compass to draw a perfect circle on the blueprint.
Protractor
Used in educational and engineering contexts.
In geometry class, everyone had a ruler and a protractor for the exercise.
Compass
Consists of two legs hinged together.
The compass legs adjusted smoothly, allowing for precise control of the radius.
Protractor
Often made of transparent material.
The transparency of the protractor helped him align it properly on the drawing.
Compass
One leg has a point, the other holds a drawing implement.
The pointed leg of the compass was anchored firmly on the paper.
Protractor
A tool used for measuring angles.
The student used a protractor to measure the angle of the triangle.
Compass
Provides dynamic measurements and constructions.
Adjusting the compass, she created several arcs for the design.
Protractor
A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Some protractors are simple half-discs or full circles.
Compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with magnetic north.
Protractor
A semicircular instrument for measuring and constructing angles.
Compass
An instrument containing a magnetized pointer which shows the direction of magnetic north and bearings from it
Walkers should be equipped with a map and compass
A magnetic compass
Crewe was ideally placed on the rail network, with connections running to all points of the compass
Protractor
(Anatomy) A muscle that extends a limb or other part.
Compass
An instrument for drawing circles and arcs and measuring distances between points, consisting of two arms linked by a movable joint, one arm ending in a point and the other usually carrying a pencil or pen
A regular heptagon cannot be constructed accurately with only ruler and compass
Protractor
One who, or that which, protracts, or causes protraction.
Compass
The range or scope of something
The event had political repercussions which are beyond the compass of this book
Goods and services which fall within the compass of the free market
Protractor
A circular or semicircular tool for drawing or measuring angles.
Compass
Go round (something) in a circular course
The ship wherein Magellan compassed the world
Protractor
(surgery) An instrument formerly used in extracting foreign or offensive matter from a wound.
Compass
Contrive to accomplish (something)
He compassed his end only by the exercise of violence
Protractor
(anatomy) A muscle that extends an organ or part; opposed to retractor.
Compass
A device used to determine geographic direction, usually consisting of a magnetic needle or needles horizontally mounted or suspended and free to pivot until aligned with the earth's magnetic field.
Protractor
An adjustable pattern used by tailors.
Compass
Another device, such as a radio compass or a gyrocompass, used for determining geographic direction.
Protractor
One who, or that which, protracts, or causes protraction.
Compass
A V-shaped device for describing circles or circular arcs and for taking measurements, consisting of a pair of rigid, end-hinged legs, one of which is equipped with a pen, pencil, or other marker and the other with a sharp point providing a pivot about which the drawing leg is turned. Also called pair of compasses.
Protractor
A mathematical instrument for laying down and measuring angles on paper, used in drawing or in plotting. It is of various forms, semicircular, rectangular, or circular.
Compass
Awareness or understanding of one's purpose or objectives
"Lacking a coherent intellectual and moral commitment, [he] was forced to find his compass in personal experience" (Doris Kearns Goodwin).
Protractor
An instrument formerly used in extracting foreign or offensive matter from a wound.
Compass
An enclosing line or boundary; a circumference
Outside the compass of the fence.
Protractor
A muscle which extends an organ or part; - opposed to retractor.
Compass
A restricted space or area
Four huge crates within the compass of the elevator.
Protractor
An adjustable pattern used by tailors.
Compass
Range or scope, as of understanding, perception, or authority
The subject falls outside the compass of this study.
Protractor
Drafting instrument used to draw or measure angles
Compass
(Music) See range.
Compass
To make a circuit of; circle
The sailboat compassed the island.
Compass
To surround; encircle
The trees compass the grave.
Compass
To understand; comprehend
"God ... is too great a profundity to be compassed by human cerebration" (Flann O'Brian).
Compass
To accomplish or bring about
"He compassed his end only by the exercise of gentle violence" (Henry James).
Compass
To gain or achieve
"She had compassed the high felicity of seeing the two men beautifully take to each another" (Henry James).
Compass
To scheme; plot
Compass the death of the king.
Compass
Forming a curve.
Compass
A magnetic or electronic device used to determine the cardinal directions (usually magnetic or true north).
Compass
A pair of compasses (a device used to draw an arc or circle).
Compass
(music) The range of notes of a musical instrument or voice.
Compass
(obsolete) A space within limits; an area.
Compass
(obsolete) An enclosing limit; a boundary, a circumference.
Within the compass of an encircling wall
Compass
Moderate bounds, limits of truth; moderation; due limits; used with within.
Compass
(archaic) Scope.
Compass
(obsolete) Range, reach.
Compass
(obsolete) A passing round; circuit; circuitous course.
Compass
To surround; to encircle; to environ; to stretch round.
Compass
To go about or round entirely; to traverse.
Compass
(dated) To accomplish; to reach; to achieve; to obtain.
Compass
(dated) To plot; to scheme (against someone).
Compass
(obsolete) In a circuit; round about.
Compass
A passing round; circuit; circuitous course.
They fetched a compass of seven day's journey.
This day I breathed first; time is come round,And where I did begin, there shall I end;My life is run his compass.
Compass
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
Compass
An inclosed space; an area; extent.
Their wisdom . . . lies in a very narrow compass.
Compass
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
The compass of his argument.
Compass
Moderate bounds, limits of truth; moderation; due limits; - used with within.
In two hundred years before (I speak within compass), no such commission had been executed.
Compass
The range of notes, or tones, within the capacity of a voice or instrument.
You would sound me from my lowest note to the top of my compass.
Compass
An instrument for determining directions upon the earth's surface by means of a magnetized bar or needle turning freely upon a pivot and pointing in a northerly and southerly direction.
He that first discovered the use of the compass did more for the supplying and increase of useful commodities than those who built workhouses | 677.169 | 1 |
The Pythagorean Theorem relates the lengths of the legs of a right triangle and the hypotenuse.
Theorem \(\PageIndex{1}\): The Pythagorean Theorem
If \(a\) and \(b\) are the lengths of the legs of the right triangle and \(c\) is the length of the hypotenuse (the side opposite the right angle) as seen in this figure.
then
\[a^{2}+b^{2}=c^{2}\nonumber\]
Proof
You can verify the truth of the Pythagorean Theorem by measuring various triangles and checking to see that they satisfy \(a^{2}+b^{2}=c^{2}\) where \(c\) is the length of the hypotenuse and \(a\) and \(b\) are the lengths of the legs. However, you would have to measure every such triangle to know its absolute truth which, of course, is impossible. We therefore would like to find a way to see it is true for all right triangles. The Pythagorean Theorem can be seen to be true by considering the following construction. Start with the triangle (any right triangle can be oriented this way)
then construct three other triangles which are the same size and shape as the first (i.e., congruent to the first): | 677.169 | 1 |
Equilateral Pentagons
Equilateral Pentagons E5 are polygons with five sides of equal
length. There are an infinite number of distinct E5 since the
internal angles can take practically any real value. Next java applet shows some
familiar geometric shapes disected into E5s. Where the disection
is not unique, an animation shows some possibilities. A selector permits to switch
between all the figures.
Theory
Only four intersected equal circles are enough to define a E5.
The center of each circle corresponds to one vertex of E5
namely angles e, a, b and c.
The fifth vertex d is determined by the intersection of circles
C1 and C4 which gives the smaller angle.
Only two angles are enough to determine the positions of the four circles, so
E5 can be defined as a two-variable-function:
E5 = F(a, b)
Conditions for Simplification
In order to avoid reflections, translations, rotations (and complications)
of a given E5 two conditions can be stated:
d must be the smaller of the five angles.
(d <= a) && (d <= b) && (d <= c) && (d <= e)
a must be greater or equal than b.
a >= b
In the following figures we can see how only one E5 is valid
among six variations. Figure 1 (in cyan) satisfies both conditions. Figures 2-5
(in green) doesn't satisfy first condition while Figure 6 (in yellow) satisfies
first conditions but don't satisfy the second one.
Covex and Concave E5
The angle c's values determine the convexity of E5.
We can conclude that:
E5 is CONVEX if angle c is lower than 180o.
E5 is CONCAVE if angle c is greater than 180o.
Self-Intersecting for E5
The angles a and b values determine the self-intersecting
of E5. We can conclude that:
E5 is not self-intersected if: 2a + b is greater than 180o.
E5 is self-intersected if: 2a + b is lower than 180o.
Some Trigonometry of E5
To find the values of angles c, d and e
as functions of (a, b) some calculations and the
following figure can help:
Here E5 is divided into three triangles. Two triangles
(left and right) are isoceles. The triangle at the center is more general.
Some angles have being divided too in this way:
a = a1 + a2 c = c1 + c2 + c4 e = e1 + e2
By trigonometry from yellow triange we found that:
a2 = c1 = p/2 - b/2 a1 = a - p/2 + b/2 B = 2 * sin(b/2)
From purple triangle we found that:
D2 = 1 + B2 - (2)(1)(B) * cos(a + b/2 - p/2)
or
D2 = 1 + 4*sin2(b/2) - 4*sin(b/2)*sin(a + b/2)
Now, having D, the angle d value can be calculated
from green triangle:
d = cos-1 [ cos(a) + cos(b) - cos(a + b) - 1/2 ]
With more calculations (too complicated to be included here) we found angle e is:
e = tan-1 [
sin(a + b) - sin(a)
-------------------------
cos(a) - cos(a + b) - 1
] + 1/2 * (p - d)
Finally angle c value is (yours better equation is welcome) is:
c = (5/2)*p - a - b - d - e
Minimums and Maximums of the angles
Having three angles c, d and e as functions
of other two (a,b), I have drawn a lot of beatiful
families of curves, they are still drawn on real paper. I hope soon have time to
include them here as electronic media... Also, other information such as area,
angles' relations can be plotted too... Analysing angles plots we get some results:
For No-self-intersected E5 these are the ranges for the
angles:
Minumum
Maximum
a
60o
120o
b
28.9o
108o
c
108o
331o
d
0o
108o
e
75.5o
240o
Equilateral Polygons' To-Do
There is a lot of material (original to be included, and not original to be
mentioned) next to be added as time permits...
Polyominoes are in fact equilateral polygons (The pentominoes are
equilateral dodecagons E12). These dodecagons have only few internal
angles values: 0o, 90o, 180o, 270o and
360o. These discrete angles surely simplifyes in some way the families
of equilateral polygons.
A Math verse reads ...every conic curve can be determined by 5 points...
Therefore exits a conic curve touching the 5 vertices of every E5.
I'm writing a Java applet for drawing conics touching all the general
E5 vertices. The relations between conic's and E5's
parameters should be interesting.
I made some calculations for equilateral hexagons (though complicated).
Three angles are needed for the hexagons. While E5 needs surface plots,
E6 needs three-dimensional plots. Altough, more conditions can be
imposed to the angles to get a reduced set. For instance, families of symmetric
equilateral hexagons having their angles values of the form
2p / n, being n an integer, tile the plane chaotically.
Chaotic Tiling with Equilateral Polygons
The rhombi are equilateral tetragons or E4. Only one angle is needed
for their description. Exists one pair of rhombi for chaotic tesselation. Also
exist one triplet of rhombi for chaotic tesselation, and one quartet, one quintet
and so on...
The CHAOS TYLES
are two equilateral pentagons
Their angles are
(80o, 160o, 60o, 140o, 100o)
and
(40o, 200o, 60o, 100o, 140o).
Applying the notation mentioned above we have this tiles are
E5(100o, 80o)
and
E5(100o, 60o).
Two symmetric equilateral hexagons I found in 1997 by drawing regular decagons,
seems can tile chaotically the plane. Their shapes recall us the main two types
of optics lenses: | 677.169 | 1 |
While playing a game of darts with your friend, you decide to see if you can plot the coordinates of where your darts land. The dartboard looks like this
Figure \(\PageIndex{1}\)
While trying to set up a rectangular coordinate system, your friend tells you that it would be easier to plot the positions of your darts using a ''polar coordinate system''. Can you do this?
Plots of Polar Coordinates
The graph paper that you have used for plotting points and sketching graphs has been rectangular grid paper. All points were plotted in a rectangular form \((x,y)\) by referring to a set of perpendicular \(x\)− and \(y\)− axes. In this section you will discover an alternative to graphing on rectangular grid paper – graphing on circular grid paper.
Look at the two options below:
Figure \(\PageIndex{2}\)Figure \(\PageIndex{3}\)
You are all familiar with the rectangular grid paper shown above. However, the circular paper lends itself to new discoveries. The paper consists of a series of concentric circles-circles that share a common center. The common center \(O\), is known as the pole or origin and the polar axis is the horizontal line \(r\) that is drawn from the pole in a positive direction. The point \(P\) that is plotted is described as a directed distance \(r\) from the pole and by the angle that \(\overline{OP} \) makes with the polar axis. The coordinates of \(P\) are \((r,\theta )\).
These coordinates are the result of assuming that the angle is rotated counterclockwise. If the angle were rotated clockwise then the coordinates of \(P\) would be \((r,−\theta )\). These values for \(P\) are called polar coordinates and are of the form \(P(r,\theta )\) where \(r\) is the absolute value of the distance from the pole to \(P\) and \(\theta \) is the angle formed by the polar axis and the terminal arm \(\overline{OP} \).
Plotting Points
Plot the point \(A(5,−255^{\circ})\) and the point \(B(3,60^{\circ})\).
To plot A, move from the pole to the circle that has \(r=5\) and then rotate \(255^{\circ}\) clockwise from the polar axis and plot the point on the circle. Label it \(A\).
Figure \(\PageIndex{4}\)
To plot B, move from the pole to the circle that has \(r=3\) and then rotate \(60^{\circ}\) counter clockwise from the polar axis and plot the point on the circle. Label it \(B\).
Figure \(\PageIndex{5}\)
Determining Pairs of Polar Coordinates
Determine four pairs of polar coordinates that represent the following point \(P(r,\theta )\) such that \(−360^{\circ}\leq \theta \leq 360^{\circ}\).
Plotting Polar Coordinates
Plot the following coordinates in polar form and give their description in polar terms: \((1,0)\), \((0,1)\), \((-1,0)\), \((-1,1)\).
Figure \(\PageIndex{7}\)
The points plotted are shown above. Since each point is 1 unit away from the origin, we know that the radius of each point in polar form will be equal to 1.
The first point lies on the positive 'x' axis, so the angle in polar coordinates is \(0^{\circ}\). The second point lies on the positive 'y' axis, so the angle in polar coordinates is \(90^{\circ}\). The third point lies on the negative 'x' axis, so the angle in polar coordinates is \(180^{\circ}\). The fourth point lies on the negative 'y' axis, so the angle in polar coordinates is \(270^{\circ}\).
Example \(\PageIndex{1}\)
Earlier, you were asked to plot the positions of your darts using a polar coordinate system.
Since you have the positions of the darts on the board with both the distance from the origin and the angle they make with the horizontal, you can describe them using polar coordinates.
Figure \(\PageIndex{8}\)
Solution
As you can see, the positions of the darts are:
\((3,45^{\circ} )\), \((6,90^{\circ} )\)
and
\((4,0^{\circ} )\)
Example \(\PageIndex{2}\)
Plot the point \(M\left(2.5, 210^{\circ} \right)\).
Solution
Figure \(\PageIndex{9}\)
Example \(\PageIndex{3}\)
Plot the point \(S\left(−3.5,\dfrac{5\pi }{6}\right)\).
Solution
Figure \(\PageIndex{10}\)
Example \(\PageIndex{4}\)
Plot the point \(A\left(1, \dfrac{3\pi }{4}\right)\).
Solution
Figure \(\PageIndex{11}\)
Review
Plot the following points on a polar coordinate grid.
\((3,150^{\circ})\)
\((2,90^{\circ})\)
\((5,60^{\circ})\)
\((4,120^{\circ})\)
\((3,210^{\circ})\)
\((−2,120^{\circ})\)
\((4,−90^{\circ})\)
\((−5,−30^{\circ})\)
\((2,−150^{\circ})\)
\((−3,300^{\circ})\)
Give three alternate sets of coordinates for the given point within the range \(−360^{\circ}\leq \theta \leq 360^{\circ}\).
\((3,60^{\circ})\)
\((2,210^{\circ})\)
\((4,330^{\circ})\)
Find the length of the arc between the points \((2,30^{\circ})\) and \((2,90^{\circ})\).
Find the area of the sector created by the origin and the points \((4,30^{\circ})\) and \((4,90^{\circ})\).
Review (Answers)
To see the Review answers, open this PDF file and look for section 6 | 677.169 | 1 |
Ans. To find the area of a triangle using determinants, you need to first find the coordinates of the three vertices of the triangle. Then, you can use the following formula: Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|, where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices.
2. What is a determinant?
Ans. In linear algebra, a determinant is a scalar value that can be calculated from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible or singular. Determinants are commonly used in various mathematical calculations, including finding the area of a triangle.
3. Why use determinants to find the area of a triangle?
Ans. Determinants are used to find the area of a triangle because they provide a convenient and efficient method to calculate the area using the coordinates of the triangle's vertices. By using determinants, you can avoid the need for lengthy calculations involving trigonometry or other geometric formulas.
4. Can determinants be used for any type of triangle?
Ans. Yes, determinants can be used to find the area of any type of triangle, whether it is scalene, isosceles, or equilateral. The method remains the same – calculating the determinant using the coordinates of the triangle's vertices – regardless of the triangle's shape or size.
5. What are some other applications of determinants in geometry?
Ans. Determinants have various applications in geometry. They can be used to find the volume of a parallelepiped, determine if three points are collinear, solve systems of linear equations, and calculate the equation of a plane. Determinants play a fundamental role in many geometric calculations and are widely used in mathematics and physics.
Text Transcript from Video
hello friends this video on determinants part 19 is brought you by exam viacom no more fear from exam before watching this video please make sure that you have watched part 1 to part 18 now we will study application of determine to find area of triangle so if you have a triangle with vertices x1 y1 x2 y2 x3 y3 so 3 vertices are threes in that case we can find the area of triangle to be 1 by 2 x1 y2 minus y3 plus y2 x2 into y 3 minus y1 plus x3 into bio minus YT the same thing can be represented in the form of determinant area of triangle is nothing but 1 by 2 into x1 y1 x2 y2 x3 y3 and you put 1 1 1 here because this is defined only for square matrix so we have to make it 3 draws 3 and that's why we put 1 1 1 here and thus we get area of triangle to be 1 by 2 X 1 y 1 1 X 2 y 2 1 X 3 y 3 1 please note this formula it's a critical for me I will be using this area of triangle to be 1 by 2 determinant of x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 and please note the area of triangle is always positive because the area is always positive so we always take absolute value for determinants so this is nothing but absolute value of this even if comes out to be negative we take the positive part absolute well we'll take some example for area of triangle let's try to find the area of the triangle with the vertices 1 0 6 0 4 3 so here if you see we have this triangle with vertices 1 0 6 0 and 14 and the area of triangle formula is very simple 1 by 2 into X 1 Y 1 1 x2 y2 1 x3 y3 1 now what is x1 y1 the first virtus 1 0 x2 y2 second what is 6 0 and x3 y3 is 4 so this is my area of prime that we just saw this so if you saw this this comes out to be 1 into 0 minus 3 minus 0 plus 1 into 6 into 3 18 minus 0 this is nothing but 1 by 2 into 0 minus 3 is minus 3 minus 3 plus 18 is 15 1 by 2 into 15 that is 15 by 2 squaring it very simple we have the points x1 y1 x2 y2 x3 y3 this put the points in the formula find it the minute get the answer and take one more example because we'll have to a very critical topic to find the area so let's find area another triangle with the vertices minus 2 minus 3 3 2 minus 1 and a same thing here so this guy is x1 y1 x2 y2 this guy is x3 y3 so area formula is nothing but 1 by 2 determinant of X 1 Y 1 1 x2 y2 1 x3 y3 1 correct this guy is nothing but 1 by 2 determinant of X 1 is minus 2 y 1 is minus 3 and 1 X 2 is 3 y 2 is 2 and 1 X 3 is minus 1 y 3 is minus 8 and one just lets all this 1 by 2 minus 2 into 2 min to 1 minus minus 18 to borrow correct minus minus 3 into 3 into 1 minus 1 into 1 3 into 1 minus 1 into minus 1 correct and then plus 1 into 8 into minus 3 or minus 8 into 3 minus 2 minus 1 this comes out to be 1 by 2 written mod of minus 2 into this is 2 minus minus a 2 plus 8 10 minus 2 into 10 is minus 20 minus 3 minus minus 3 plus 3 plus 3 into 3 plus 1 for 4 in degrees 12 plus 12 this guy becomes minus a 2 3 is minus 24 plus 2 minus 20 minus windy this guy is nothing but 1 by 2 into minus 30 correct and this guy's nothing but 15 minus 15 but we take absolute volume value of a so I take the absolute value of this this becomes 15 unit or square and if you have teal square and that is my answer so this question I took because the answer came out to be minus 15 and we have to note here that the area of triangle is always positive so I will take the absolute value absolute value of minus 15 is 15 thank you visit exam viacom to watch free education videos try free online tests get the best quality study materials study from the best tutors and mentors and much more thanks again
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attachment main-qimg-5456f088c373424b9e0b766a7449d264.png is no longer available addition of a parallel line allows us to see a variety of equivalent angles (and similar triangles!)
First, let's examine these two similar triangles. Since side AD (on the blue triangle) is twice the length of side AE (on the red triangle), we can conclude that the blue triangle is twice as large as the red triangle. So, side AP is twice the length of side AF. This tells us that side PF = side AF
Now let's examine these two similar triangles. Since side BC (on the purple triangle) is twice the length of side DC (on the green triangle), we can conclude that the purple triangle is twice as large as the green triangle. So, side FC is twice the length of side PC. This tells us that side PF = side PC
At this point, we know that side PF = side AF and side PF = side PC Since both equations are set equal to PF, we can conclude that AF = FP = PC
This is a tough one. I guessed - the only progress I was able to make is that BD and DC are equal. I thought AD bisects angle A but I don't see how I can use that to help me and I'm not sure I can make that conclusion even though it makes sense given that BD = DC because each point on the bisector is equidistant from each the sides of the angle.
We also know that AE = ED because E is the midpoint of AD but I don't see how we can use that with what we know about BF to progress. Would love it if someone could provide a good explanation.
Consider a point, X, on AC somewhere between F and C. The point X should be such that DX is parallel to BF. Now, since BF || DX so, CX=XF (parallel base theorem, since BD=DC) Also, AF= FX (parallel base theorem, since AE=ED)Re: In triangle ABC, D is the mid-point of BC and E is the mid-point of AD
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10 Apr 2023, 22 triangle ABC, D is the mid-point of BC and E is the mid-point of AD [#permalink] | 677.169 | 1 |
Hemisphere Calculator
Introduction
Overview of Hemispheres
A hemisphere is a three-dimensional geometric shape that consists of half of a sphere. It is characterized by a flat base (the great circle) and a curved surface that meets at a single point, the apex.
Importance of Calculations in Geometry
Calculations in geometry, such as those involving hemispheres, are crucial for various fields including architecture, engineering, and science. They enable precise measurement and analysis of shapes and volumes, aiding in design, construction, and understanding natural phenomena.
Understanding Hemispheres
Definition of a Hemisphere
A hemisphere is a three-dimensional geometric shape that consists of half of a sphere. It is bounded by a flat circular base (known as the great circle) and a curved surface that extends from the base to a single point, the apex.
Real-World Examples
Hemispheres are commonly found in various natural and man-made objects:
The Earth's atmosphere is approximately divided into two hemispheres: northern and southern. | 677.169 | 1 |
How many trigonometric formulas are there in class 11?
How many trigonometric formulas are there in class 11?
In Mathematics, there are a total of six different types of Trigonometric functions: sine (sin), Cosine (cos), Secant (sec), Cosecant (cosec), Tangent (tan) and Cotangent (cot).
What is the formulas of trigonometry?
Hypotenuse: It is the side opposite to the right angle of the triangle (or the largest side). Accordingly, the trigonometry ratios are written as follows based on the corresponding angles: sine or sinθ = Perpendicular/Hypotenuse = Opposite/Hypotenuse. cosine or cosθ = Base/ Hypotenuse = Adjacent/Hypotenuse.
How do you find full marks in trigonometry?
11 Tips to Conquer Trigonometry Proving
Tip 1) Always Start from the More Complex Side.
Tip 2) Express everything into Sine and Cosine.
Tip 3) Combine Terms into a Single Fraction.
Tip 4) Use Pythagorean Identities to transform between sin²x and cos²x.
Tip 5) Know when to Apply Double Angle Formula (DAF)
Is trigonometry useful in life?
Trigonometry and its functions have an enormous number of uses in our daily life. For instance, it is used in geography to measure the distance between landmarks, in astronomy to measure the distance of nearby stars and also in the satellite navigation system.
How many hours should I study maths in Class 11?
Minimum hours to study – It is good to dedicate at least 6 hours to study apart from the college hours. And there is no need to study continuously. Taking sufficient intervals while studying helps retain the interest without overwhelming the mind.
How many formulas are there in trigonometry?
six
The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent. By using a right-angled triangle as a reference, the trigonometric functions and identities are derived: sin θ = Opposite Side/Hypotenuse. cos θ = Adjacent Side/Hypotenuse.
What is the rhyme for trigonometry?
SOH-CAH-TOA Sine = Opposite ÷ Hypotenuse.
What are all the formulas of trigonometry?
Trigonometric Ratios
Trigonometric Ratios for Unit Circle. Similarly,for a unit circle,for which radius is equal to 1,and θ is the angle.
Trigonometry Identities
Even and Odd Angle Formulas
Co-function Formulas
Double Angle Formulas
Half Angle Formulas
Thrice of Angle Formulas
Sum and Difference Formulas
Product to Sum Formulas
What are the six trigonometry functions?
The six main trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. They are useful for finding heights and distances, and have practical applications in many fields including architecture, surveying, and engineering.
What is the formula for trigonometry?
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4×3 − 3x + d = 0, where x is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
What are trig identities?
The Pythagorean formula for sines and cosines. This is probably the most important trig identity.
Identities for negative angles. Sine,tangent,cotangent,and cosecant are odd functions while cosine and secant are even functions.
Double angle formulas for sine and cosine. Note that there are three forms for the double angle formula for cosine. | 677.169 | 1 |
A vector has magnitude same as that of $$\vec{A}=3 \hat{i}+4 \hat{j}$$ and is parallel to $$\vec{B}=4 \hat{i}+3 \hat{j}$$. The $$x$$ and $$y$$ components of this vector in first quadrant are $$x$$ and 3 respectively where $$x=$$ _________.
Vectors $$a\widehat i + b\widehat j + \widehat k$$ and $$2\widehat i - 3\widehat j + 4\widehat k$$ are perpendicular to each other when $$3a + 2b = 7$$, the ratio of $$a$$ to $$b$$ is $${x \over 2}$$. The value of $$x$$ is ____________.
Your input ____
4
JEE Main 2022 (Online) 28th July Morning Shift
Numerical
+4
-1
If the projection of $$2 \hat{i}+4 \hat{j}-2 \hat{k}$$ on $$\hat{i}+2 \hat{j}+\alpha \hat{k}$$ is zero. Then, the value of $$\alpha$$ will be ___________. | 677.169 | 1 |
Problem 1
The 3 lines \(x=3, y-2.5=\text-\frac{1}{5}(x-0.5),\) and \(y-2.5=x-3.5\) intersect at point \(P\). Find the coordinates of \(P\). Verify algebraically that the lines all intersect at \(P\).
Solution
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Problem 2
Triangle \(ABC\) has vertices at \((0,0), (5,5),\) and \((10,1)\). Kiran calculates the point of intersection of the medians using the following steps:
Draw the triangle.
Calculate the midpoint of each side.
Draw the medians.
Write an equation for 2 of the medians.
Solve the system of equations.
Use Kiran's method to calculate the point of intersection of the medians.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 16.)
Problem 3
Triangle \(ABC\) and its medians are shown. Write an equation for median \(AE\).
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 16.)
Problem 4
Given \(A=(1,2)\) and \(B=(7,14)\), find the point that partitions segment \(AB\) in a \(2:1\) ratio.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 15.)
Problem 5
A quadrilateral has vertices \(A=(0,0), B=(4,6), C=(0,12),\) and \(D=(\text-4,6)\). Mai thinks the quadrilateral is a rhombus and Elena thinks the quadrilateral is a square. Do you agree with either of them? Show or explain your reasoning.
Solution
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(From Unit 6, Lesson 14.)
Problem 6
The image shows a graph of the parabola with focus \((\text-3,\text-2)\) and directrix \(y=2\), and the line given by \(y=\text-3\). Find and verify the points where the parabola and the line intersect.
Solution
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(From Unit 6, Lesson 13.)
Problem 7
For each equation, is the graph of the equation parallel to the line shown, perpendicular to the line shown, or neither?
\(y=0.25x\)
\(y=2x – 4\)
\(y-2 = \text-4(x-3)\)
\(2y + 8x = 7\)
\(x-4y=3\)
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 12.)
Problem 8
Write 2 equivalent equations for a line with \(x\)-intercept \((3,0)\) and \(y\)-intercept \((0, 2)\).
Solution
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(From Unit 6, Lesson 9.)
Problem 9
Parabola A and parabola B both have the line \(y=\text-2\) as the directrix. Parabola A has its focus at \((3, 4)\) and parabola B has its focus at \((5,0)\). Select all true statements.
Parabola A is wider than parabola B.
Parabola B is wider than parabola A.
The parabolas have the same line of symmetry.
The line of symmetry of parabola A is to the right of that of parabola B.
The line of symmetry of parabola B is to the right of that of parabola A.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution. | 677.169 | 1 |
How do you change 240 degrees to radians?
How do you change 240 degrees to radians?
To convert degrees to radians, multiply by π180° π 180 ° , since a full circle is 360° or 2π radians. Cancel the common factor of 60 . Factor 60 60 out of 240 240 .
What is the angle of 240 degree?
The reference angle of 240° is 60°.
How many degrees is a 1/4 revolution?
Comparing Revolutions, Degrees, and Radians
words
rev
deg
quarter turn
1/4
90°
half turn
1/2
180°
three-quarter turn
3/4
270°
full turn
1
360°
What is the reference angle for 14pi 3 explain your thinking?
π3
143π−2π≡83π≡83π−2π≡23π . Because they are coterminal, the reference angle for 143π is the same. It is π3 .
What is Rev and rad?
Amount: 1 revolution (rev) of angle. Equals: 6.28 radians (rad) in angle. Converting revolution to radians value in the angle units scale. TOGGLE : from radians into revolutions in the other way around.
How many radians are in a revolution circle?
6.28 radians
Radian measure does not have to be expressed in multiples of . Remember that , π ≈ 3.14 , so one complete revolution is about 6.28 radians, and one-quarter revolution is , 1 4 ( 2 π ) = π 2 , or about 1.57 radians.
What is the formula for converting degrees to radians?
The equation used to convert degrees to radians is rad = (deg x pi)/180, where rad stands for radians, deg is degrees and pi is equal to 3.142. Radians and degrees are ways to measure an angle in a circle.
1) Write down the number of degrees you want to convert to radians. Let's work with a few examples so you really get the concept down. 2) Multiply the number of degrees by π/180. To understand why you have to do this, you should know that 180 degrees constitute π radians. 3) Do the math. Simply carry out the multiplication process, by multiplying the number of degrees by π/180. 4) Now, you've got to put each fraction in lowest terms to get your final answer. 5) Write down your answer. To be clear, you can write down what your original angle measure became when converted to radians.
How do you convert radian measures to degrees?
From the latter, we obtain the equation 1 radian = ( 180 π ) o . This leads us to the rule to convert radian measure to degree measure. To convert from radians to degrees, multiply the radians by 180 ° π radians . | 677.169 | 1 |
Articles relating to circles, a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. | 677.169 | 1 |
Question 1.
Make two halves. Show different ways. Shade half of each rectangle.
Answer:
Equal halves for the given rectangle is drawn and each half of each rectangle is shaded as shown in the figure above.
Different Shapes of a Third of the Same Rectangle
Question 2.
Make three thirds. Show different ways. Shade a third of each rectangle.
Answer:
Equal thirds for the given rectangle is drawn and each third of each rectangle is shaded as shown in the figure above.
Different Shapes of a Fourth of the Same Rectangle
Question 3.
Make four-fourths. Show different ways. Shade a fourth of each rectangle.
Answer:
Equal four-fourths for the given rectangle is drawn and each fourth of each rectangle is shaded as shown in the figure above.
Equal Shares Using the Same Square
Question 4.
Make 2 equal shares. Show different ways. Shade half of each square.
Answer:
Equal halves for the given square is drawn and each half of each square is shaded as shown in the figure above.
Question 5.
Make 3 equal shares. Show different ways. Shade a third of each square.
Answer:
Equal thirds for the given square is drawn and each third of each square is shaded as shown in the figure above.
Question 6.
Make 4 equal shares. Show different ways. Shade a fourth of each square.
Answer:
Equal four-fourths for the given square is drawn and each fourth of each square is shaded as shown in the figure above.
Equal Shares Using the Same Circle
Question 7.
Make 2 equal shares. Shade half of the circle.
Answer:
2 equal shares of the given circle is drawn and one half of the circle is shaded.
Question 8.
Make 3 equal shares. Shade a third of the circle.
Answer:
3 equal shares of the given circle is drawn and a third of the circle is shaded.
Question 9.
Make 4 equal shares. Shade a fourth of the circle.
Answer:
4 equal shares of the given circle is drawn and a fourth of the circle is shaded.
Different Shape but Same Size
Question 10.
Use Drawings 1, 2, and 3 to explain why the blue and yellow shares are equal.
Answer:
In figure 1, 4 equal shares are drawn for the given rectangle.
In figure 2, 8 equal shares are drawn for the given rectangle.Out of 8 parts, 2 are shaded which equals to 1 part of the figure 1.
That is the reason why the blue and yellow shares are equal. | 677.169 | 1 |
Orthogonal Matrix Calculator – Quick and Precise
To use this calculator, input a 3×3 matrix into the provided text area. Separate the elements of each row by a comma and each row by a newline.
Click the "Calculate" button to determine if the matrix is orthogonal. The result will be displayed in the result input field.
How it works:
The calculator checks if the matrix is orthogonal by first transposing the matrix. Next, it multiplies the original matrix with the transposed one. Finally, it checks if the resultant matrix is an identity matrix. If the resultant matrix is an identity matrix, the original matrix is orthogonal.
Limitations:
This calculator only handles 3×3 matrices. The input must be numeric and correctly formatted for the calculator to work. Invalid inputs will generate an error message indicating that the input is invalid.
Use Cases for This Calculator
Calculate Determinant of an Orthogonal Matrix
Enter the values of a 3×3 orthogonal matrix and instantly find the determinant. Obtain the result in a single click to determine the overall scaling factor of the matrix, which indicates how it affects the area or volume it operates on.
Check if Matrix is Orthogonal
Input any square matrix to verify if it is orthogonal using the calculator. Confirm if the matrix columns are orthonormal vectors to ensure that the matrix has a valid rotation or reflection operation.
Find Inverse of an Orthogonal Matrix
Submit the orthogonal matrix values and obtain the inverse matrix immediately. Use this feature to efficiently compute reflections or rotations, reversing the transformation operation of the original matrix.
Calculate Transpose of an Orthogonal Matrix
Explore the tool to quickly transpose an orthogonal matrix. Swap the rows and columns to see the effect on the matrix; this enables you to analyze reflections, rotations, or other transformations more effectively.
Compute Eigenvalues of an Orthogonal Matrix
Enter the matrix elements to determine the eigenvalues directly with ease. Analyze the stability of the matrix transformations by understanding how the values scale the eigenvectors during the transformation process.
Calculate Trace of an Orthogonal Matrix
Input a square matrix and instantly get the trace value. Utilize this calculation to verify properties of the matrix or understand how the elements affect the main diagonal of the matrix.
Verify if Matrix is Idempotent
Check any square matrix to see if it is idempotent. Use the calculator to confirm if multiplying the matrix by itself results in the original matrix, assisting in analyzing projection or reflection operations.
Find Rank of an Orthogonal Matrix
Input a matrix to compute the rank value promptly. Employ this function to determine the dimension of the output space affected by the matrix transformation, serving insights into the impact of the transformation.
Compute Null Space of an Orthogonal Matrix
Insert the matrix values to calculate the null space efficiently. Understand which vectors are unaffected by the transformation operation of the matrix, providing insights into properties like rotation or reflection symmetry.
Calculate Spectral Decomposition of an Orthogonal Matrix
Submit the matrix elements to instantly obtain the spectral decomposition. Break down the matrix into eigenvalues and eigenvectors to simplify complex transformations and understand the impact of the matrix on different dimensions. | 677.169 | 1 |
A Pythagorean tree is generated by taking a square and adding 2 smaller squares that have been rotated and dilated so a right-angle triangle is formed. The hypotenuse of the right-angled triangle is the length of the original square and the other legs of the triangle are the lengths of the added squares. By continuing this with each successive square created, we can create the Pythagorean tree fractal. For example, here are the first few iterations when the added squares have equal lengths: (when we create an isosceles right triangle between the 3 squares and the two added squares are congruent). For this investigation, we will focus only on Pythagorean tree fractals created with equally sized squares at each stage so that isosceles right-angled triangles are created in between the squares. the use of TI Nspire CAS is incorporated in the investigation. The extension includes walks with binary numbers. | 677.169 | 1 |
Hint: Here in this question belongs to construction topic, we have to construct the perpendicular line at point P on the line AB of length 6.2 cm, where point P lies 4cm from the point B in the line AB by using a geometrical instruments like centimeter scale, compass with provision of fitting a pencil and protractor.
Complete step by step solution: Perpendicular lines are defined as two lines that meet or intersect each other at right angles (\[{90^ \circ }\]). Now, consider the given question Draw a line segment \[AB = 6.2\] cm Mark a point \[P\], in \[AB\] such that \[BP = 4\] cm through point \[P\] draw perpendicular to \[AB\]. To construct the perpendicular line follow the below steps:. Steps of Construction: I.Draw a line segment \[AB = 6.2\] cm II.Take point 'B' as centre and radius 4cm construct an arc 'P' on line segment AB. III.With P as centre and some radius draw arc meeting AB at the points C and D. IV.With C, D as centres and equal radii [each is more than half of CD] draw two arcs, meeting each other at the point O. V.Join OP. Then OP is perpendicular for line AB and it makes an angle \[{90^ \circ }\]. The construction of perpendicular line is
Note: When doing construction handling the instruments carefully, remember when making an arc between an angle the radius will be the same which cannot be alter and perpendicular is a line that makes an angle of \[{90^ \circ }\] with another line. \[{90^ \circ }\] is also called a right angle and is marked by a little square between two perpendicular lines otherwise the two lines intersect at a right angle, and hence, are said to be perpendicular to each other. If we can verify the construction easily by measuring the angle using an instrument called a protractor. | 677.169 | 1 |
Mathematical figure measured in degrees Crossword Clue
General knowledge plays a crucial role in solving crosswords, especially the Mathematical figure measured in degrees crossword clue which has appeared on May 23 2024 Crosswords with Friends puzzle. The answer we have shared for Mathematical figure measured in degreesL
E
Definition
•
The inclosed space near the point where two lines meet; a
corner; a nook.
•
The figure made by. two lines which meet.
•
The difference of direction of two lines. In the lines meet,
the point of meeting is the vertex of the angle.
•
A projecting or sharp corner; an angular fragment.
•
A name given to four of the twelve astrological "houses."
•
A fishhook; tackle for catching fish, consisting of a line,
hook, and bait, with or without a rod.
•
To fish with an angle (fishhook), or with hook and line.
•
To use some bait or artifice; to intrigue; to scheme; as,
to angle for praise.
•
To try to gain by some insinuating artifice; to allure.
Listed here below we have the other possible answers to Mathematical figure measured in degrees crossword clue.
Rank
Answer
Clue
Publisher
99%
ANGLE
Mathematical figure measured in degrees
Crosswords with Friends
Recent Usage in Crossword Puzzles:
Crosswords with Friends, May 23 2024
The most accurate solution to Mathematical figure measured in degrees crossword clue is ANGLE
There are a total of 5 letters in Mathematical figure measured in degrees crossword clue
The Mathematical figure measured in degrees crossword clue based on our database appeared on May 23 2024 Crosswords with Friends puzzle | 677.169 | 1 |
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Fifth postulate
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Euclid's axiom of parallelism
One can draw just one straight line through a point $P$ not on a straight line $AA_1$ that does not intersect $AA_1$ and lies in the plane containing $P$ and $AA_1$. In Euclid's Elements the fifth postulate is given in the following equivalent form: "If a straight line incident to two straight lines has interior angles on the same side of less than two right angles, then the extension of these two lines meets on that side where the angles are less than two right angles" (see [1]). Among the commentators of Euclid there arose the view that a proof of this statement could be found based on the remaining axioms. Attempts at proofs occurred as long ago as in Ancient Greece. These attempts continued in the East in the Middle Ages and then in Western Europe. If direct logical mistakes are overlooked, then usually an implicit (and sometimes also a clearly understood) assumption was made which was not deducible from the remaining axioms and which turned out to be equivalent to the fifth postulate. For example, the distance between parallels is bounded, the space admits a "simple" motion (all trajectories are straight lines), two converging straight lines always intersect, there exist similar but unequal figures, the sum of the angles in a triangle is equal to two right angles, etc. G. Saccheri (1733) considered a quadrangle with right angles at the base and with equal lateral sides. Omar Khayyam (11th–12th century) had considered such a quadrangle earlier. Of the three possible hypotheses about the remaining two equal angles (they are obtuse, they are acute, they are right angles) he tried to reject the first two since the third implied the fifth postulate. Saccheri succeeded in deriving a contradiction from the first hypothesis, but he made a logical mistake in refuting the hypothesis on the acute angle. J. Lambert (1766, published 1786) with a similar approach refuted the hypothesis on the acute angle but also made a serious mistake. He assumed that such a geometry is realized only on an imaginary sphere. A. Legendre (1800), in the first edition of the textbook Eléments de la géométrie, started from the sum $S$ of the angles of a triangle. Having rejected the hypothesis $S>\pi$, he made a mistake in deriving the consequences of the hypothesis $S<\pi$, namely, he implicitly introduced the axiom that for any point inside an acutely-angled sector there exists a straight line passing through this point and intersecting both sides of the sector. The solution to the problem of the fifth postulate (more precisely its removal) was obtained by a geometry created by N.I. Lobachevskii (1826) in which the fifth postulate does not hold. From the fact that Lobachevskii geometry is consistent, it follows that the fifth postulate is independent of the other axioms in Euclidean geometry. | 677.169 | 1 |
Q. Consider a triangle drawn on the X – Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is (CAT 2005)
Q. Two sides of rhombus ABCD are parallel to the lines y = x + 2 and y = 7x + 3. If the diagonal of rhombus intersect at the point (1, 2) and the vertex A lies on y-axis find the possible co-ordinates of A.
Q. Find the equation of the straight line which passes through the point of intersection of the straight lines x + y = 8 and 3x - 2y + 1 = 0 and is parallel to the straight line joining the points (3, 4) and (5, 6). | 677.169 | 1 |
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All › Mixed Age Year 4 and 5 Properties of Shape Step 1 Free Resource Pack
Mixed Age Year 4 and 5 Properties of Shape Step 1 Free Resource Pack
Step 1: Mixed Age Year 4 and 5 Properties of Shape Step 1
Mixed Age Year 4 and 5 Properties of Shape Step 1 Resource Pack includes a teaching PowerPoint and differentiated varied fluency and reasoning and problem solving resources for this step which covers Year 4 Identify Angles & Year 5 Measuring Angles in Degrees for Summer Block 4.
What's included in the Pack?
This Mixed Age Year 4 and 5 Properties of Shape Step 1 pack includes:
Mixed Age Year 4 and 5 Properties of Shape Step 1 Teaching PowerPoint with examples.
Year 4 Identify Angles Varied Fluency with answers.
Year 4 Identify Angles Reasoning and Problem Solving with answers.
Year 5 Measuring Angles in Degrees Varied Fluency with answers.
Year 5 Measuring Angles in Degrees Reasoning and Problem Solving with answers.
Varied Fluency Developing Questions to support identifying acute, obtuse and right angles. Angles in horizontal plane and facing one direction. Angles obviously visually different. Angle tester used as pictorial support. Expected Questions to support identifying acute, obtuse and right angles. Most angles in horizontal plane and facing in any direction. Angles visually similar. Angle tester used as pictorial support in some questions. Greater Depth Questions to support identifying acute, obtuse and right angles. Angles in any plane and facing any direction. Includes some intersecting lines with multiple angles.
Reasoning and Problem Solving Questions 1, 4 and 7 (Problem Solving) Developing Combine angles to find which remain acute. Two digit values with 5 or 0 in the units, angles combine 1:1. Expected Combine angles to find which remain acute. Any two digit values addition crossing tens, some combine 2:1 and some have no match. Greater Depth Using a given angle, calculate which angle they can add to create the largest acute angle and smallest obtuse angle.
Questions 2, 5 and 8 (Reasoning) Developing Spotting the odd one out, obvious differences all angles presented on a horizontal plane and facing one direction. Angle tester used as pictorial support. Expected Spotting the odd one out, subtle differences, all angles presented on a horizontal plane, facing any direction with one confusing factor. Angle tester used as pictorial support for some questions. Greater Depth Spotting the odd one out, very subtle differences, angles presented on a variety of planes and facing any direction. with a number of confusing factors.
Questions 3, 6 and 9 (Problem Solving) Developing Using 4 digit cards, create either obtuse or acute angles. Expected Using 4 digit cards, decide whether can make more acute or obtuse angles. Greater Depth Using 4 digit cards, decide whether can make more acute or obtuse angles, some double digits.
Differentiation for Year 5 Measuring Angles in Degrees:
Varied Fluency Developing Questions to support measuring degrees around a point, including angles in increments of 90°. Using right angles and reflex angles. Clock faces and compasses used in quarter increments. Expected Questions to support measuring degrees around a point, including angles in increments of 30o and 45°. Using acute, obtuse, reflex and right angles. Clock faces used in increments of twelve and compasses used in increments of eight. Greater Depth Questions to support measuring degrees around a point, including some angles in increments of 30° and 45°. Using acute, obtuse, reflex and right angles. Clock faces used in increments of twelve and compasses used in increments of eight, where some or no increments are marked.
Reasoning and Problem Solving Questions 1, 4 and 7 (Reasoning) Developing Explain which statement is correct when describing the turn needed to face a compass point, including angles in increments of 90°. Using right angles and reflex angles. Expected Explain which statement is correct when describing the turn needed to face a compass point, including angles in increments of 30o and 45°. Using acute, right angle, obtuse and reflex angles. Greater Depth Explain which statement is correct when describing the turns needed to face a compass point, including one or more angles in increments of 30o and 45°. Using acute, right angle, obtuse and reflex angles, no pictorial support.
Questions 2, 5 and 8 (Problem Solving) Developing Find all of the possible options to make more than/less than statements true. Including angles in increments of 90°. Expected Find all of the possible options to make more than/less than statements true. Including angles in increments of 30o and 45°. Greater Depth Find all of the possible options to make more than/less than statement true. Including one or more angles in increments of 30o and 45°. Including more than one whole and a mixture of angles and fractions.
Questions 3, 6 and 9 (Problem Solving) Developing Find all of possible times that a clock face could show after a sequence of quarter turns clockwise. Times to the nearest fifteen minutes. Expected Find all of possible times that a clock face could show after a sequence of quarter turns clockwise or anti-clockwise. Times to the nearest 5 minutes. Greater Depth Find all of possible times that a clock face could show after a sequence of twelfth turns clockwise or anti-clockwise. Times to the nearest minute | 677.169 | 1 |
Computational Geometry
Computational geometry is a branch of computer science and mathematics that deals with the study of algorithms and methods used to solve geometric problems by using computers. It involves understanding the properties and relationships of geometric shapes and objects in two or three dimensions and developing efficient algorithms to manipulate and analyze them. This specialized skill is used in various fields, including computer graphics, robotics, computer-aided design (CAD), and geographic information systems (GIS). | 677.169 | 1 |
Formula
Calculating Secant of an Angle
.
The secant of an angle \( \theta \) is defined as:
\( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \)
where:
hypotenuse is the length of the hypotenuse (the side opposite the right angle).
adjacent is the length of the side adjacent to angle \( \theta \).
Example 1
Let's calculate the secant secant of 30 degrees is: secant of 45 degrees is:
These examples demonstrate how to calculate the secant of an angle manually, showing the importance of understanding the trigonometric function secant in various mathematical and practical applications.
{
"topic": "sec",
"function": "sec",
"function_desc": "Secant",
"x_symbol": "θ",
"x_desc": "angle",
"category": "Trigonometry",
"formula": "\\( sec(θ) = \\frac{Length~ of~ Hypotenuse}{Length~ of~ Adjacent~ side} \\)",
"formula_in_js": "1/Math.cos(θ)",
"description": "In a right angled triangle, Secant of an angle is the ratio of the length of the hypotenuse to the length of the adjacent side.",
"template": "trigonometry_function",
"content": "<h2>Calculating Secant of an Angle</h2>\n<p>.</p>\n<p>The secant of an angle \\( \\theta \\) is defined as:</p>\n<p>\\( \\sec(\\theta) = \\frac{\\text{hypotenuse}}{\\text{adjacent}} \\)</p>\n<p>where:</p>\n<ul>\n<li><b>hypotenuse</b> is the length of the hypotenuse (the side opposite the right angle).</li>\n<li><b>adjacent</b> is the length of the side adjacent to angle \\( \\theta \\).</li>\n</ul>\n<h3>Example 1</h3>\n<p>Let's calculate the secant secant of 30 degrees is:</p>\n<p>\\( \\sec(30^{\\circ}) = \\frac{\\text{hypotenuse}}{\\text{adjacent}} = \\frac{1}{\\cos(30^{\\circ})} = \\frac{1}{0.866} \\approx 1.154 secant of 45 degrees is:</p>\n<p>\\( \\sec(45^{\\circ}) = \\frac{\\text{hypotenuse}}{\\text{adjacent}} = \\frac{1}{\\cos(45^{\\circ})} = \\frac{1}{\\frac{\\sqrt{2}}{2}} = \\sqrt{2} \\approx 1.414 \\)</p>\n<p>These examples demonstrate how to calculate the secant of an angle manually, showing the importance of understanding the trigonometric function secant in various mathematical and practical applications.</p>"
} | 677.169 | 1 |
What is the unit vector that is normal to the plane containing <1,1,1> and <2,0,-1>?
1 Answer
Explanation:
You must do the cross product of the two vectors to obtain a vector perpendicular to the plane:
The cross product is the deteminant of #∣((veci,vecj,veck),(1,1,1),(2,0,-1))∣#
#=veci(-1)-vecj(-1-2)+veck(-2)=〈-1,3,-2〉#
We check by doing the dot products. #〈-1,3,-2〉.〈1,1,1〉=-1+3-2=0# #〈-1,3,-2〉.〈2,0,-1〉=-2+0+2=0#
As the dots products are #=0#, we conclude that the vector is perpendicular to the plane. #∥vecv∥=sqrt(1+9+4)=sqrt14#
The unit vector is #hatv=vecv/(∥vecv∥)=1/sqrt14〈-1,3,-2〉# | 677.169 | 1 |
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. Elements of Geometry and Conic Sections - Page 10 by Elias Loomis - 1849 - 226 pages Full view - About this book
...BA is perpendicular to AT and Definition I.10, which states that "When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the right angles is right, and the straight line standing on the other is called a perpendicular to...
...classes of (rectilinear) angles (trans. Heath 1926, I, 181): 1 0. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right [...]. 1 1 . An obtuse angle is an angle greater than a right angle. 12....
...is equal to the angle EHC. [1.8] And they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is 30 called a perpendicular...
...straight line on which it stands.' The full translation then is: 'When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to...
...definition, it is not necessary to define a rectilineal angle). 10. SJt When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to...
...EB ; therefore the angle AFE is equal to the angle BFE. [i. 8] But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right ; [i. Def. 10] 20 therefore each of the angles AFE, BFE is right. Therefore... | 677.169 | 1 |
Geometric Coloring Pages Free Printable
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You can go with shades of single color, for smooth transitions and depth, or you can use a rainbow of hues for a vibrant and energetic look. Web geometric coloring pages can be as diverse as your imagination allows. Search through 109300 colorings, dot to dots, tutorials and silhouettes. Home / coloring pages / geometric. Ideal for math enthusiasts and.
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Web 3d cubes, spheres, pyramids, and other three dimensional figures; Ideal for math enthusiasts and pattern lovers alike, this. You can go with shades of single color, for smooth transitions and depth, or you can use a rainbow of hues for a vibrant and energetic look. Web geometric coloring pages can be as diverse as your imagination allows. Web geometric.
Web geometric coloring pages | free printable pictures. Web get ready to explore the mesmerizing world of shapes and angles with these 20 geometric coloring pages, free for you to download and print! Search through 109300 colorings, dot to dots, tutorials and silhouettes. Web geometric coloring pages can be as diverse as your imagination allows. Web 3d cubes, spheres, pyramids, and other three dimensional figures; This geometric coloring page features triangles that are. Home / coloring pages / geometric. Web print these geometric coloring pages for free and have fun with it. Ideal for math enthusiasts and pattern lovers alike, this. You can go with shades of single color, for smooth transitions and depth, or you can use a rainbow of hues for a vibrant and energetic look. And brick, swirled, mosaic, or tiled patterns. Scroll down for for a wonderland of geometric shapes and designs! You can also check out our other shapes coloring pages on our website.
Web Get Ready To Explore The Mesmerizing World Of Shapes And Angles With These 20 Geometric Coloring Pages, Free For You To Download And Print!
Web geometric coloring pages | free printable pictures. This geometric coloring page features triangles that are. You can also check out our other shapes coloring pages on our website. Search through 109300 colorings, dot to dots, tutorials and silhouettes.
Ideal For Math Enthusiasts And Pattern Lovers Alike, This.
Web geometric coloring pages can be as diverse as your imagination allows. Home / coloring pages / geometric. Web print these geometric coloring pages for free and have fun with it. You can go with shades of single color, for smooth transitions and depth, or you can use a rainbow of hues for a vibrant and energetic look. | 677.169 | 1 |
Welcome to the study plan for the Class 6 Mathematics chapter on "Understanding Elementary Shapes." In this chapter, you will delve into essential concepts, including "Measuring Line Segments," "Angles – 'Right' and 'Straight'," "Angles – 'Acute', 'Obtuse' and 'Reflex'," and "Measuring Angles." By the end of this guide, you'll have a clear strategy for effective learning and excelling in this chapter as well as in your Class 6 Mathematics exam.
Follow Your Timetable with EduRev!
The plan we've created for this chapter will help you make the best use of your time simply click here.
Topics to Cover
Before we jump into the study plan, let's go through the topics we need to cover in this chapter:
Measuring Line Segments
Angles – 'Right' and 'Straight'
Angles – 'Acute', 'Obtuse' and 'Reflex'
Measuring Angles
Simple Strategies for Parents to Help Kids in Studying
Parents play a crucial role in supporting their children's learning journey. Here's how parents can help their children study the chapter "Understanding Elementary Shapes":
Engage in discussions about the chapter's content and progress with your child.
Revision
On the revision day, consolidate your knowledge by revisiting the key concepts related to measuring line segments, angles, and angle measurement. Test yourself by solving questions from the chapter test, worksheets, and NCERT Exemplar questions.
Feedback and Questions
If you have any questions or doubts about the chapter "Understanding Elementary ShapesFollow this study plan diligently to master the concepts of "Understanding Elementary Shapes" and perform exceptionally in your Class 6 Mathematics exam. Certainly! Here are all the links mentioned in the study plan, organized under suitable headers: Study Plan and Resources:
Document Description: 3 Days Timetable: Understanding Elementary Shapes for Class 6 2024 is part of Chapter-wise Time Table for Class 6 preparation.
The notes and questions for 3 Days Timetable: Understanding Elementary Shapes have been prepared according to the Class 6 exam syllabus. Information about 3 Days Timetable: Understanding Elementary Shapes covers topics
like Simple Strategies for Parents to Help Kids in Studying, Timetable for "Understanding Elementary Shapes", Day 1: Measuring Line Segments, Day 2: Angles – 'Right' and 'Straight', Day 3: Angles – 'Acute', 'Obtuse' and 'Reflex'; Measuring Angles, Revision and 3 Days Timetable: Understanding Elementary Shapes Example, for Class 6 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for 3 Days Timetable: Understanding Elementary Shapes.
Introduction of 3 Days Timetable: Understanding Elementary Shapes in English is available as part of our Chapter-wise Time Table for Class 6
for Class 6 & 3 Days Timetable: Understanding Elementary Shapes in Hindi for Chapter-wise Time Table for Class 6 course.
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Exam by signing up for free. Class 6: 3 Days Timetable: Understanding Elementary Shapes | Chapter-wise Time Table for Class 6
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The 3 Days Timetable: Understanding Elementary Shapes Understanding Elementary Shapes now and kickstart your journey towards success in the Class 6 exam.
Importance of 3 Days Timetable: Understanding Elementary Shapes
The importance of 3 Days Timetable: Understanding Elementary Shapes Understanding Elementary Shapes Notes
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3 Days Timetable: Understanding Elementary Shapes Class 6 Questions
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Students of Class 6 can study 3 Days Timetable: Understanding Elementary Shapes alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the 3 Days Timetable: Understanding Elementary Shapes Understanding Elementary Shapes is prepared as per the latest Class 6 syllabus. | 677.169 | 1 |
I - 07 EIGHT 07 EIGHT
Key elements:
straight line intersection
equal proportions
"In" is called when the kite is at ^90<25.
After 25% horizontal flight the kite enters an 'eight' figure.
The diameter of each arc is approximately 30% with the bottom of the eight at ^10.
The point where the two straight lines intersect is at ^50.
The kite "leaves" the 'eight' figure at ^90<>0, the same point where it entered.
"Out" is called when the kite has flown 25% horizontally at ^90. | 677.169 | 1 |
find the missing angle of a quadrilateral worksheet pdf
Finding The Missing Angle Worksheet Pdf … Read more
Find The Missing Angles Of A Quadrilateral … Read more
Find The Missing Angle Worksheet
How To Find The Missing Angle Of A Quadrilateral … Read more
Quadrilateral Find The Missing Angle | 677.169 | 1 |
how to find the missing angle of an isosceles triangle | 677.169 | 1 |
Medians the line segments joining the vertices to the mid-points of the opposite side of a triangle are known as its medians.
Video Solution
Text Solution
Verified by Experts
Median
You are familiar with the term 'triangle' and its properties. Before we try to explore more about them, we need to get some good information on some of their basic properties. We know that a three-sided polygon which has 3 vertices and 3 sides enclosing 3 angles is called atriangle.
A triangle can be classified intoscalene, isoscelesandequilateralbased on the length of its sides. On the basis of interior angles, a triangle can be an obtuse-angled, acute-angled, or right-angled triangle. We also need to know that the sum of interior angles in a triangle is 180 degrees.
So, in this chapter, we will learn how to draw medians of a triangle. We will also concentrate on the properties of medians of a triangle as well to get the perfect idea about it.
What is the Median of a Triangle?
When a line segment joins a vertex to the mid-point of the side and that too it is opposite to that vertex, it is known as a median of the triangle. Look at the below figure where the median AD divides BC into two equal halves, i.e., DB = DC.
Properties of Median of a Triangle
Coming to the properties of median of a triangle, they are as follows:
There are 3 medians in every triangle, one from each vertex. The 3 medians of the triangle ABC are AE, BF and CD.
It is always in a single point that the 3 medians irrespective of the shape of the triangle is.
The triangle is cut into two smaller triangles by each median of a triangle which have equal areas.
The point where the 3 medians meet is known as the centroid of the triangle. The centroid of the triangle ABC is point O.
The triangle is divided into 6 smaller triangles of equal area by the 3 medians.
How to Find the Median of a Triangle?
If a, b and c are considered to be the lengths of the sides and m is the median from interior angle A to side a, then we can find the median of a triangle as: | 677.169 | 1 |
Which angle is Coterminal with 112?
Some of the coterminal angles of 112° are 472°, 832°, -248°, and -608°.
How do you find the positive and negative Coterminal angles?
For example, −330°and 390°are all coterminal. To find a positive and a negative angle coterminal with a given angle, you can add and subtract 360°if the angle is measured in degrees or 2π if the angle is measured in radians.
What is the Coterminal angle of 330?
Coterminal angle of 330° (11π / 6): 690°, 1050°, -30°, -390°
What is the Coterminal angle of 90?
The coterminal angles of 90° are 450°, 810°, -270°, and -630°.
What is the Coterminal angle of 30 degrees?
Coterminal angles: are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side. For example, the angles 30°, –330° and 390° are all coterminal (see figure 2.1 below).
Which angle is Coterminal to a 175 angle?
Illustration showing coterminal angles of 175° and -185°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.
How do you find the least positive coterminal angle?
Do you want to find a coterminal angle of a given angle,preferably in the[,360°) range?
Are you hunting for positive and negative coterminal angles?
Would you like to check if two angles are coterminal?
Are you searching for a coterminal angles calculator for radians?
Or maybe you're looking for a coterminal angles definition,with some examples?
How to solve coterminal angles and reference angles?
The difference (in any order) of any two coterminal angles is a multiple of 360°
To find the coterminal angle of an angle,we just add or subtract multiples of 360°.
The number of coterminal angles of an angle is infinite because there is an infinite number of multiples of 360°.
What does coterminal angle mean?
C degrees larger or smaller than the other. | 677.169 | 1 |
JEE Main 2022 June 27 – Shift 1 Maths Question Paper with Solutions
The JEE Main 2022 June 27 – Shift 1 Maths Question Paper with Solutions are provided on this page. These solutions are prepared by experts at BYJU'S. Students are recommended to study the JEE Main 2022 question paper with solutions so that they can be familiar with the type of questions asked for JEE Main 2022. They can use the JEE Main 2022 answer keys to calculate their score in the JEE Main 2022 exams. Students can download the JEE Main 2022 June 27 – Shift 1 Maths Question Paper with Solutions in PDF format.
SECTION – A
Multiple Choice Questions: This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The area of the polygon, whose vertices are the non-real roots of the equation
12. In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If (α, β) is the centroid of ΔABC, then 15(α + β) is equal to :
Numerical Value Type Questions: This section contains 10 questions. In Section B, attempt any five questions out of 10. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 06.25, 07.00, –00.33, –00.30, 30.27, –27.30) using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. | 677.169 | 1 |
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Cartesian coordinates
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Cartesian coordinates
Cartesian coordinateskärtēˈzhən [key] [for René Descartes], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers (x,y) representing the distances of the point from two intersecting straight lines, referred to as the x-axis and the y-axis. The point of intersection of these axes, which are called the coordinate axes, is known as the origin. In rectangular coordinates, the type most often used, the axes are taken to be perpendicular, with the x-axis horizontal and the y-axis vertical, so that the x-coordinate, or abscissa, of P is measured along the horizontal perpendicular from P to the y-axis (i.e., parallel to the x-axis) and the y-coordinate, or ordinate, is measured along the vertical perpendicular from P to the x-axis (parallel to the y-axis). In oblique coordinates the axes are not perpendicular; the abscissa of P is measured along a parallel to the x-axis, and the ordinate is measured along a parallel to the y-axis, but neither of these parallels is perpendicular to the other coordinate axis as in rectangular coordinates. Similarly, a point in space may be specified by the triple of numbers (x,y,z) representing the distances from three planes determined by three intersecting straight lines not all in the same plane; i.e., the x-coordinate represents the distance from the yz-plane measured along a parallel to the x-axis, the y-coordinate represents the distance from the xz-plane measured along a parallel to the y-axis, and the z-coordinate represents the distance from the xy-plane measured along a parallel to the z-axis (the axes are usually taken to be mutually perpendicular). Analogous systems may be defined for describing points in abstract spaces of four or more dimensions. Many of the curves studied in classical geometry can be described as the set of points (x,y) that satisfy some equation f(x,y)=0. In this way certain questions in geometry can be transformed into questions about numbers and resolved by means of analytic geometry | 677.169 | 1 |
find the measure of each numbered angle answer key | 677.169 | 1 |
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venn diagram formula for 4 sets
Use of Venn-Diagrams in Set Theory. We will discuss below representing data using the method of Venn diagrams for 2 groups and 3 groups: First, From the above figure, consider the following data: The box denotes a class having N students. Venn Diagram Formula for 2 sets / circles: Where, A only = A - (A∩B) B only = B - (A∩B) Related Calculator: First, let's look at this example: This Venn diagram shows overlapping sets that contain letters. This page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could do for $<4$ sets. [Venn diagram. The N students are divided as below: Labels have been simplified for greater readability." In a Venn-diagram, the universal set U is described by a point enclosed in a rectangle while its subsets are described by points within the closed curves (generally circles) inside the rectangle. However, as with two sets, we'll write in an easier-to-remember notation, in just a moment. "Five-set Venn diagram using congruent ellipses in a radially symmetrical arrangement devised by Branko Grünbaum. Venn diagrams are the mathematical diagrams, which are drawn to illustrate a clear connection between sets. Wikipedia] This 5-set Venn diagram template for the ConceptDraw PRO diagramming and vector drawing software is included in the Venn Diagrams solution from the area "What is a Diagram" of ConceptDraw Solution Park. This is a 4 Circle Venn Diagram template. On this example you can see the intersections of 4 sets A, B, C and D. Use MyDraw to create your own 4 set Venn diagram in minutes. Here's how it works: the circle represents all the elements in a given set while the areas of intersection characterize the elements that simultaneously belong to multiple sets. Venn Diagram Formula for 2 sets / circles: Where, A only = A - (A∩B) B only = B - (A∩B) Related Calculator: Three overlapping sets can be depicted as follows: And the overlap formula is. A Venn diagram is a chart that compares two or more sets (collections of data) and illustrates the differences and commonalities between them with overlapping circles. Part 4 Union And Intersection Formula Of Venn Diagrams With 7 Inclusionexclusion Principle Wikipedia ... Combinatorics Why Can A Venn Diagram For 4 Sets Not Be 68 Best Venn Diagram Templates Images In 2019 Venn Diagram Form 4 Maths Sets Bimbingan Matematik Uncle Zul Venn Diagram/Overlapping Sets. Venn diagrams is a convenient way of representing data. | 677.169 | 1 |
$$ABC$$ is a triangle. $$D$$ is the middle point of $$BC$$. If $$AD$$ is perpendicular to $$AC$$, then prove that
$$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$$
2
IIT-JEE 1980
Subjective
+3
-0
$$ABC$$ is a triangle with $$AB=AC$$. $$D$$ is any point on the side $$BC$$. $$E$$ and $$F$$ are points on the side $$AB$$ and $$AC$$, respectively, such that $$DE$$ is parallel to $$AC$$, and $$DF$$ is parallel to $$AB$$. Prove that
$$$DF + FA + AE + ED = AB + AC$$$
3
IIT-JEE 1980
Subjective
+5
-0
(i) $$PQ$$ is a vertical tower. $$P$$ is the foot and $$Q$$ is the top of the tower. $$A, B, C$$ are three points in the horizontal plane through $$P$$. The angles of elevation of $$Q$$ from $$A$$, $$B$$, $$C$$ are equal, and each is equal to $$\theta $$. The sides of the triangle $$ABC$$ are $$a, b, c$$; and the area of the triangle $$ABC$$ is $$\Delta $$. Show that the height of the tower is $${{abc\tan \theta } \over {4\Delta }}$$.
(ii) $$AB$$ is vertical pole. The end $$A$$ is on the level ground. $$C$$ is the middle point of $$AB$$. $$P$$ is a point on the level ground. The portion $$CB$$ subtends an angle $$\beta $$ at $$P$$. If $$AP = n\,AB,$$ then show that tan$$\beta $$ $$ = {n \over {2{n^2} + 1}}$$
4
IIT-JEE 1979
Subjective
+5
-0
(a) A balloon is observed simultaneously from three points $$A, B$$ and $$C$$ on a straight road directly beneath it. The angular elevation at $$B$$ is twice that at $$A$$ and the angular elevation at $$C$$ is thrice that at $$A$$. If the distance between $$A$$ and $$B$$ is a and the distance between $$B$$ and $$C$$ is $$b$$, find the height of the balloon in terms of $$a$$ and $$b$$.
(b) Find the area of the smaller part of a disc of radius $$10$$ cm, cut off by a chord $$AB$$ which subtends an angle of at the circumference. | 677.169 | 1 |
...the supplement of the arc FAH£, or the jingle FCB is the supplement of the angle Fc3. (K) The tine of an arc is a straight line drawn from one end of that arc, perpendicular to a diameter passing through the other end of the same arc. Thus FG is the...
...Thus, the arc BL is the supplement of the arc AB, and the angle BCL of the angle ACB c. 9. THE CHORD of an arc is a straight line drawn from one end of the arc to the other. b In lite manner AB is the complement of BE, and the angle ACB of the angle BCE. The...
...... " , i, . ....... :• ' •,• .!I 81. To facilitate the calculations in trigonometry, there *re drawn, within and about the circle, a number of straight...drawn from the end G of the arc, perpendicular to the diameter AM which passes through the other end "A of the arc. Cor. The sine is half the chord of double...
...equiangular and .-. equilateral. The chord of a quadrant =.R/v/2. (17.) DEF. 6. The Sine or Rig/it Sine of an arc is a straight line drawn from one end of the arc perpendicular to the diameter passing through the other end of the arc. . Thus, if from P, PE be drawn perpendicular...
...circle. For the triangle AOP is equiangular and .'. equilateral. The chord of a quadrant = (17.) DEF. 6. a straight line drawn from one end of the arc perpendicular to the diameter passing through the other end of the arc. Thus, if from P, PE be drawn perpendicular to...
...equiangular and .•. equilateral. The chord of a quadrant = -R^/2. (17.) DEF. 6. The Sine or Right Sine of an arc is a straight line drawn from one end of the arc perpendicular to the diameter passing through the other end of the arc. Thus, if from P, PE be drawn perpendicular to...
...remainder between that arc and a semi-circle. Thus, the arc given being AB, its supplement is BC. 229. The SINE OF AN ARC is a straight line, drawn from one extremity of the arc, upon and perpendicular to a radius or diameter. M Thus, BM is the sine of the...
...arc BF is the supplement of the arc FAH¿>, or the angle FCB is the supplement of the angle FcZ>. (K) The sine of an arc is a straight line drawn from one end of that arc, perpendicular to a diameter passing through the other end of the same arc. Thus FG is the...
...drawn, within and about the circle, a number of straight lines, called Sines, Tangents, Secants, Sfc. With these the learner should make himself perfectly...drawn from the end G of the arc, perpendicular to the diameter AM which passes through the other end A of the arc. Cor. The sine is half the chord of double...
...of straight lines, called Sines, Tangents, Secants, fyr. With these the learner should make himseif perfectly familiar. 82. The SINE of an arc is a straight...end of the arc, perpendicular to a diameter which pastes through the other end. Thus BG (Fig. 3.) is the sine of the arc AG. For BG is a line drawn from... | 677.169 | 1 |
The MCQ Quiz: Operation of drawing representative line between two vectors in such a way that tail of one vector coincides with the head of other vector is called; "Applied Physics: Vectors" App (Play Store & App Store) with answers: Vector subtraction; Vector addition; Vector division; Vector multiplication; for ACT test prep classes. Practice Vector and Equilibrium Questions and Answers, Apple Book to download free sample for two year online colleges. | 677.169 | 1 |
Question 10.
Show that the given points form a parallelogram:
A (2.5,3.5), B(10, -4), C(2.5, -2.5) and D(-5, 5).
Solution:
Let A(2.5, 3.5), B(10, -4), C(2.5, -2.5) and D(-5, 5) are the vertices of a parallelogram.
Slope of AB = Slope of CD = -1
∴ AB is Parallel to CD ……(1)
Slope of BC = Slope of AD
∴ BC is parallel to AD
From (1) and (2) we get ABCD is a parallelogram. | 677.169 | 1 |
2 Answers
If the missing side, c, is not the hypotenuse
then b, the longer side must be the hypotenuse, and
by the Pythagorean Theorem #b^2 = c^2+ a^2#
or #c^2 = b^2-a^2# #= 12^2-5^2# #= 119# #rarr c = sqrt(119)# | 677.169 | 1 |
Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra
Scalar triple product
The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar.
Scalar triple product
Definition 6.4
For a given set of three vectors , , and , the scalar (× ) ⋅ is called a scalar triple product of , , .
Remark
Note
Given any three vectors , , and c the following are scalar triple products:
Geometrical interpretation of scalar triple product
Geometrically, the absolute value of the scalar triple product ( ×) .. is the volume of the parallelepiped formed by using the three vectors , and as co-terminus edges. Indeed, the magnitude of the vector (× ) is the area of the parallelogram formed by using and ; and the direction of the vector (× ) is perpendicular to the plane parallel to both and .
Therefore, | (× ) ⋅| is | × || || cosθ | , where θ is the angle between × and . FromFig. 6.17, we observe that | | | cosθ | is theheight of the parallelepiped formed by using thethree vectors as adjacent vectors. Thus, | (× ) ⋅| is the volume of the parallelepiped.
The following theorem is useful for computing scalar triple products.
Theorem 6.1
Proof
By definition, we have
which completes the proof of the theorem.
Properties of the scalar triple product
Theorem 6.2
For any three vectors , and , (× ) ⋅= ⋅ (× ) .
Proof
Hence the theorem is proved.
Note
By Theorem 6.2, it follows that, in a scalar triple product, dot and cross can be interchanged without altering the order of occurrences of the vectors, by placing the parentheses in such a way that dot lies outside the parentheses, and cross lies between the vectors inside the parentheses. For instance, we have
Notation
For any three vectors , and , the scalar triple product (× ) ⋅ is denoted by [× , ] . [× , ] is read as box a, b, c . For this reason and also because the absolute value of a scalar triple product represents the volume of a box (rectangular parallelepiped),a scalar triple product is also called a box product.
Note
(1)
In other words, [, , ] = [, , ] = [, , ] ; that is, if the three vectors are permuted in the same cyclic order, the value of the scalar triple product remains the same.
(2) If any two vectors are interchanged in their position in a scalar triple product, then the value of the scalar triple product is (-1) times the original value. More explicitly,
We have studied about coplanar vectors in XI standard as three nonzero vectors of which, one can be expressed as a linear combination of the other two. Now we use scalar triple product for the characterisation of coplanar vectors.
Theorem 6.4
The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar.
Proof
Let , , be any three non-zero vectors. Then,
× ⋅= 0 ⇔is perpendicular to ×
⇔lies in the plane which is parallel to both and
, , are coplanar.
Theorem 6.5
Three vectors , , are coplanar if, and only if, there exist scalars r, s, t ∈R such that atleast one of them is non-zero and r+s+t= . | 677.169 | 1 |
Question 5.
Take some round bowls and plates. Place their edges one upon the other to find pairs of congruent edges. (Textbook pg. no. 10)
Solution:
[Students should attempt the above activities on their own.] | 677.169 | 1 |
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How to Use a Surveyor's Theodolite
Ron Fasick
•
Updated December 17, 2018
Petr Makeev/iStock/GettyImages
A surveyor's theodolite is used to measure horizontal and vertical angles. These measurements are used to plot boundary lines, building foundations and utility routing. A theodolite measures distance manually by chains of standardized length or metal measuring tapes along the length of the desired angle. Land-surveying students use a theodolite as a way to learn the principles of angle measurement. Surveying on the job requires the use of more-advanced surveying instruments.
Drive a surveyor's nail into the ground at the point where you want to set up the theodolite. Angles will be measured from this point, as will distances.
Set up the tripod legs, taking care to set the height where the instrument sight will be at a comfortable eye level. Be certain to check that the hole in the center of the mounting plate is located over the nail. Press each leg into the ground by stepping on the bracket at the bottom of each leg.
Fine-tune the position of the legs so that the mounting plate on the top of the tripod is as eye-level as possible.
Remove the theodolite from the case. Most theodolites have a sturdy handle on top. This is the best place to lift the instrument. Gently place the instrument on the mounting plate and screw in the mounting knob beneath the instrument.
Adjust the theodolite to level by adjusting the tripod legs, using the bull's-eye level. Fine-tune the adjustment with the leveling knobs on the instrument.
Adjust the small sight, called the vertical plummet, at the bottom of the theodolite. This sight allows you to make certain the instrument is centered directly over the nail. Fine-tune the vertical plummet by adjusting the knobs on the bottom of the theodolite.
Look through the main scope and aim the crosshairs at the point to be measured. Twist the locking knobs to hold the theodolite in position on the exact point. View the horizontal and vertical angles in the viewing scope on the side of the instrument.
Tip
To fully understand the use of surveying instruments, take a course from a licensed surveyor. The hands-on experience is invaluable in understanding the more abstract concepts associated with mathematics and instrument operation.
Warning
Use care when transporting and operating the theodolite. These precise measuring devices are very sensitive, and they are expensive to repair if damaged.
To fully understand the use of surveying instruments, take a course from a licensed surveyor. The hands-on experience is invaluable in understanding the more abstract concepts associated with mathematics and instrument operation.
Warnings
Use care when transporting and operating the theodolite. These precise measuring devices are very sensitive, and they are expensive to repair if damaged.
Writer
From South Carolina, Ron Fasick began writing in 2001 for local music news website BluegrassJam.net. Following his experience there, Fasick spent several years in the construction and engineering sectors, where he garnered strong technical and report writing skills. Fasick holds an Associate of Science in civil engineering technology from Central Piedmont Community College and studied music at Winthrop University. | 677.169 | 1 |
RD Sharma Solutions for Class 8 Chapter 19 Visualising Shapes
1. What is the least number of planes that can enclose a solid? What is the name of the solid?
Solution:
The least number of planes that are required to enclose a solid is 4.
The name of solid is tetrahedron.
2. Can a polyhedron have for its faces? (i) 3 triangles? (ii) 4 triangles? (iii) a square and four triangles?
Solution:
(i) 3 triangles?
No, because a polyhedron is a solid shape bounded by polygons.
(ii) 4 triangles?
Yes, because a tetrahedron as 4 triangles as its faces.
(iii) a square and four triangles?
Yes, because a square pyramid has a square and four triangles as its faces.
3. Is it possible to have a polyhedron with any given number of faces?
Solution:
Yes, if number of faces is four or more.
4. Is a square prism same as a cube?
Solution:
Yes. We know that a square is a three dimensional shape with six rectangular shaped sides, out of which two are squares. Cubes are of rectangular prism length, width and height of same measurement.
5. Can a polyhedron have 10 faces, 20 edges and 15 vertices?
Solution:
No.
Let us use Euler's formula
V + F = E + 2
15 + 10 = 20 + 2
25 ≠ 22
Since the given polyhedron is not following Euler's formula, therefore it is not possible to have 10 faces, 20 edges and 15 vertices.
6. Verify Euler's formula for each of the following polyhedrons:
Solution:
(i) Vertices = 10
Faces = 7
Edges = 15
By using Euler's formula
V + F = E + 2
10 + 7 = 15 + 2
17 = 17
Hence verified.
(ii(iii) Vertices = 14
Faces = 8
Edges = 20
By using Euler's formula
V + F = E + 2
14 + 8 = 20 + 2
22 = 22
Hence verified.
(iv) Vertices = 6
Faces = 8
Edges = 12
By using Euler's formula
V + F = E + 2
6 + 8 = 12 + 2
14 = 14
Hence verified.
(v7. Using Euler's formula find the unknown:
Faces
?
5
20
Vertices
6
?
12
Edges
12
9
?
Solution:
(i)
By using Euler's formula
V + F = E + 2
6 + F = 12 + 2
F = 14 – 6
F = 8
∴ Number of faces is 8
(ii)
By using Euler's formula
V + F = E + 2
V + 5 = 9 + 2
V = 11 – 5
V = 6
∴ Number of vertices is 6
(iii)
By using Euler's formula
V + F = E + 2
12 + 20 = E + 2
E = 32 – 2
E = 30
∴ Number of edges is 30
EXERCISE 19.2 PAGE NO: 19.12
1. Which among of the following are nets for a cube?
Solution:
Figure (iv), (v), (vi) are the nets for a cube.
2. Name the polyhedron that can be made by folding each net:
Solution:
(i) From figure (i), a Square pyramid can be made by folding each net.
(ii) From figure (ii), a Triangular prism can be made by folding each net.
(iii) From figure (iii), a Triangular prism can be made by folding each net.
(iv) From figure (iv), a Hexagonal prism can be made by folding each net.
(iv) From figure (v), a Hexagonal pyramid can be made by folding each net.
(v) From figure (vi), a Cube can be made by folding each net.
3. Dice are cubes where the numbers on the opposite faces must total 7. Which of the following are dice?
Solution:
Figure (i), is a dice. Since the sum of numbers on opposite faces is 7 (3 + 4 = 7 and 6 + 1 = 7).
4. Draw nets for each of the following polyhedrons:
Solution:
(i) The net pattern for cube is
(ii) The pattern for triangular prism is
(iii) The net pattern for hexagonal prism is
(iv) The net pattern for pentagonal pyramid is
5. Match the following figures:
Solution:
(a)-(iv) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 4×4 = 16
(b)-(i) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 8×3 = 24
(c)-(ii) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 6×3 = 18
(d)-(iii) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 3×9 = 27 | 677.169 | 1 |
determine the following characteristics of the graph of a linear function, given its equation or graph:
B.4, describe the characteristics of 3-D objects and 2-D shapes, and analyse the relationships among them.
C.1: : solve problems involving two right triangles using trigonometry and the Pythagorean Theorem | 677.169 | 1 |
Two opposite sides of a parallelogram each have a length of #5 #. If one corner of the parallelogram has an angle of #(5 pi)/8 # and the parallelogram's area is #12 #, how long are the other two sides?
Given that two opposite sides of a parallelogram each have a length of 5, and one corner of the parallelogram has an angle of ( \frac{5\pi}{8} ), we can use the formula for the area of a parallelogram to find the lengths of the other two sides.
The area of a parallelogram is given by the formula ( A = ab \sin(\theta) ), where ( A ) is the area, ( a ) and ( b ) are the lengths of the sides, and ( \theta ) is the angle between the sides.
Given ( A = 12 ), ( a = b = 5 ), and ( \theta = \frac{5\pi}{8} ), we can solve for the lengths of the other two sides.
[ 12 = 5 \times 5 \times \sin\left(\frac{5\pi}{8}\right) ]
[ 12 = 25 \times \sin\left(\frac{5\pi}{8}\right) ]
[ \sin\left(\frac{5\pi}{8}\right) = \frac{12}{25} ]
Now, we can use the sine ratio to find the lengths of the other two sides. Let ( x ) be the length of one of these sides. Then, the length of the other side is also ( x | 677.169 | 1 |
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