## Pythagorean Theorem

How to Use The Pythagorean Theorem

## The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$ \overline{c}$$ is always the hypotenuse . Remember that this formula only applies to right triangles .

## Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the following examples to see pictures of the formula.

## Conceptual Animation of Pythagorean Theorem

Demonstration #1.

More on the Pythagorean theorem

## Demonstration #2

Video tutorial on how to use the pythagorean theorem.

## Step By Step Examples of Using the Pythagorean Theorem

Example 1 (solving for the hypotenuse).

Use the Pythagorean theorem to determine the length of X.

Identify the legs and the hypotenuse of the right triangle .

The legs have length 6 and 8 . $$X $$ is the hypotenuse because it is opposite the right angle.

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2 $

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X $

## Example 2 (solving for a Leg)

The legs have length 24 and $$X$$ are the legs. The hypotenuse is 26.

$ \red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2 $

$ \red x^2 + 24^2= 26^2 \\ \red x^2 + 576= 676 \\ \red x^2 = 676 - 576 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10 $

## Practice Problems

Find the length of X.

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse .

The legs have length 14 and 48 . The hypotenuse is X.

$ A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2 $

Solve for the unknown.

$ 14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x} $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest tenth.

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs .

The legs have length 9 and X . The hypotenuse is 10.

$ A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2 $

$ 9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4 $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest hundredth.

The legs have length '10' and 'X'. The hypotenuse is 20.

$ A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2 $

$ 10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32 $

- Pythagorean Theorem Shell Problem Cool challenging problem from involving pythagoren theorem, isosceles triangles and pattern recognition
- Free Triangle Worksheets
- Pythagorean Theorem Calculator

## Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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## Module 11: Geometry

Using the pythagorean theorem to solve problems, learning outcomes.

- Use the pythagorean theorem to find the unknown length of a right triangle given the two other lengths

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[/latex] BCE.

Remember that a right triangle has a [latex]90^\circ [/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\circ [/latex] angle is called the hypotenuse, and the other two sides are called the legs. See the triangles below.

In a right triangle, the side opposite the [latex]90^\circ [/latex] angle is called the hypotenuse and each of the other sides is called a leg.

## The Pythagorean Theorem

In any right triangle [latex]\Delta ABC[/latex],

[latex]{a}^{2}+{b}^{2}={c}^{2}[/latex]

where [latex]c[/latex] is the length of the hypotenuse [latex]a[/latex] and [latex]b[/latex] are the lengths of the legs.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation [latex]\sqrt{m}[/latex] and defined it in this way:

[latex]\text{If }m={n}^{2},\text{ then }\sqrt{m}=n\text{ for }n\ge 0[/latex]

For example, we found that [latex]\sqrt{25}[/latex] is [latex]5[/latex] because [latex]{5}^{2}=25[/latex].

We will use this definition of square roots to solve for the length of a side in a right triangle.

Use the Pythagorean Theorem to find the length of the hypotenuse.

Use the Pythagorean Theorem to find the length of the longer leg.

Kelvin is building a gazebo and wants to brace each corner by placing a [latex]\text{10-inch}[/latex] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

In the following video we show two more examples of how to use the Pythagorean Theorem to solve application problems.

- Question ID 146918, 146916, 146914, 146913. Authored by : Lumen Learning. License : CC BY: Attribution
- Solve Applications Using the Pythagorean Theorem (c only). Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/2P0dJxpwFMY . License : CC BY: Attribution
- Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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## Course: 8th grade > Unit 5

- Use Pythagorean theorem to find area of an isosceles triangle
- Use Pythagorean theorem to find perimeter
- Use Pythagorean theorem to find area
- Pythagorean theorem word problem: carpet
- Pythagorean theorem word problem: fishing boat

## Pythagorean theorem word problems

- Pythagorean theorem in 3D
- Your answer should be
- an integer, like 6
- an exact decimal, like 0.75
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4
- a mixed number, like 1 3 / 4

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## How to Use the Pythagorean Theorem

Last Updated: October 6, 2023 Fact Checked

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 425,294 times.

The Pythagorean Theorem describes the lengths of the sides of a right triangle in a way that is so elegant and practical that the theorem is still widely used today. The theorem states that for any right triangle, the sum of the squares of the non-hypotenuse sides is equal to the square of the hypotenuse. In other words, for a right triangle with perpendicular sides of length a and b and hypotenuse of length c, a 2 + b 2 = c 2 . The Pythagorean Theorem is one of the fundamental pillars of basic geometry, having countless practical applications - using the theorem, for instance, it's easy to find the distance between two points on a coordinate plane.

## Finding the Sides of a Right Triangle

- As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded "curve", to identify them as such. Look for this special mark in one of the corners of your triangle.

- Let's say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we're not sure what the length of the third side is. In this case, we know we're solving for the length of the third side, and, because we know the lengths of the other two, we're ready to go! We'll return to this example problem in the following steps.
- If the lengths of two of your sides are unknown, you'll need to determine the length of one more side to use the Pythagorean Theorem. Basic trigonometry functions can help you here if you know one of the non-right angles in the triangle.

- In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5²

- In our example, we would square 3 and 5 to get 9 and 25 , respectively. We can rewrite our equation as 9 + b² = 25.

- In our example, our current equation is 9 + b² = 25. To isolate b², let's subtract 9 from both sides of the equation. This leaves us with b² = 16.

- In our example, b² = 16, taking the square root of both sides gives us b = 4. Thus, we can say that the length of the unknown side of our triangle is 4 .

- a² + b² = c²
- (5)² + (20)² = c²
- 25 + 400 = c²
- sqrt(425) = c
- c = 20.6 . The approximate length of the ladder is 20.6 meters (67.6 ft) .

## Calculating the Distance Between Two Points in an X-Y Plane

- To find the distance between these two points, we will treat each point as one of the non-right angle corners of a right triangle. By doing this, it's easy to find the length of the a and b sides, then calculate c, the hypotenuse, which is the distance between the two points.

- |x 1 - x 2 |
- |y 1 - y 2 |
- So, we can say that in our right triangle, side a = 3 and side b = 4.

- (3)²+(4)²= c² c= sqrt(9+16) c= sqrt(25) c= 5. The distance between (3,5) and (6,1) is 5 .

## Community Q&A

- sqrt(x) means the "square root of x". Thanks Helpful 0 Not Helpful 0
- Another check - the longest side will be opposite the largest angle and the shortest side will be opposite the smallest angle. Thanks Helpful 0 Not Helpful 0
- across from the right angle (not touching the right angle)
- the longest side of the right triangle
- substituted for c in the Pythagorean theorem

## You Might Also Like

## Expert Interview

Thanks for reading our article! If you’d like to learn more about math, check out our in-depth interview with Grace Imson, MA .

- ↑ https://www.mathwarehouse.com/geometry/triangles/how-to-use-the-pythagorean-theorem.php
- ↑ http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U07_L1_T4_text_final.html
- ↑ https://www.mathsisfun.com/pythagoras.html
- ↑ https://www.mathplanet.com/education/pre-algebra/right-triangles-and-algebra/the-pythagorean-theorem
- ↑ https://www.cuemath.com/geometry/pythagoras-theorem/
- ↑ https://sciencing.com/real-life-uses-pythagorean-theorem-8247514.html
- ↑ https://courses.lumenlearning.com/waymakercollegealgebra/chapter/distance-in-the-plane/
- ↑ https://www.varsitytutors.com/hotmath/hotmath_help/topics/distance-formula
- ↑ https://www.mathplanet.com/education/algebra-1/radical-expressions/the-distance-and-midpoint-formulas
- ↑ https://www.cuemath.com/geometry/hypotenuse/

## About This Article

To use the Pythagorean Theorem on a triangle with a 90-degree angle, label the shorter sides of the triangle a and b, and the longer side opposite of the right angle should be labelled c. As long as you know the length of two of the sides, you can solve for the third side by using the formula a squared plus b squared equals c squared. Place your known values into the equation and solve for the unknown variable, then take the square root of both sides of the equation to get the result. If you want to use the Pythagorean Theorem to find the distance between 2 points, keep reading the article! Did this summary help you? Yes No

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## Pythagorean Theorem Calculator

Enter the values of any two angles and any one side of a triangle below for which you want to find the length of the remaining two sides.

The Pythagorean theorem calculator finds the length of the remaining two sides of a given triangle using sine law or definitions of trigonometric functions.

If a given triangle is a right angle triangle, then the length of the remaining two sides of the triangle can be calculated using the definition of sine and cosine functions given by the formula-

sin θ = Opposite Side Hypotenuse

cos θ = Adjacent Side Hypotenuse

If a given triangle is not a right angle triangle, then the length of the remaining two sides of the triangle can be calculated using the sine law given by the formula-

Where A , B and C are angles of the triangle.

Click the blue arrow to submit. Choose "Find the Length of the Third Side" from the topic selector and click to see the result in our Trigonometry Calculator!

Find the Length of the Third Side

## Popular Problems:

Find the Length of the Third Side Side Angle b = A = 90 c = B = 3 a = 4 C = Find the Length of the Third Side Side Angle b = A = 90 c = B = 1 a = 2 C = Find the Length of the Third Side Side Angle b = A = 90 c = B = 2 a = 2 C = Find the Length of the Third Side Side Angle b = A = 90 c = B = a = 4 C = 40 Find the Length of the Third Side Side Angle b = 4 A = 75 c = B = 20 a = C =

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## IMAGES

## VIDEO

## COMMENTS

Step 1 Identify the legs and the hypotenuse of the right triangle . The legs have length 6 and 8. X X is the hypotenuse because it is opposite the right angle. Step 2 Substitute values into the formula (remember 'C' is the hypotenuse). A2 + B2 = C2 62 + 82 = X2 A 2 + B 2 = C 2 6 2 + 8 2 = X 2 Step 3 Solve for the unknown.

Multi-Step Pythagorean Theorem Problems Date_____ Period____ Find the area of each triangle. Round intermediate values to the nearest tenth. Use the rounded values to calculate the next value. Round your final answer to the nearest tenth. 1) 9 8 2) 9 7 3) 6 8 4) 7 8 5) 5 5 7.6 6) 7 7 5.2 7) 12 10 8 8) 7 5 8-1-

Free worksheet at https://www.kutasoftware.com/freeige.htmlGo to ️ https://maemap.com/math/geometry/ ⬅️ for more Geometry information!Please support me: ?...

0:00 / 4:34 Multi-step word problem with Pythagorean theorem | Geometry | Khan Academy Khan Academy 8.05M subscribers Subscribe 17K views 3 years ago Use the Pythagorean theorem and...

Kuta Software - Infinite Geometry Name Period Multi-Step Pythagorean Theorem Proble s Date Find the area of each triangle. Round intermediate va ueS to the nearest tenth. Use the rounded values to calculate the next value. Round your final answer to the nearest tenth. 5.2 322 7.6 12 Z 10 10) 6.4 12) 16 9/0/3 17 11) 29 13) 25 15) 26 11 22 11 16

Explaining mult-step Pythagorean theorem problems.

Here's the Pythagorean Theorem formula for your quick reference. Problem 1: Find the value of x x in the right triangle. Problem 2: Find the value of x x in the right triangle. Problem 3: Find the value of x x in the right triangle. Problem 4: The legs of a right triangle are 5 5 and 12 12.

3 years ago Could multiplying the six and 2.5 by ten first also work? • ( 7 votes) Upvote Downvote Flag victoriamathew12345 3 years ago No, because the 6 in the triangle is not part of the length of the string instead it is the distance Lainey runs. However, multiplying the 2.5 and sqrt (42.25) by 10 would still work. They both equal to 90m.

0:00 / 5:27 How to solve for a multi-step Pythagorean theorem question: Calculator Included Swell Academy 188 subscribers Subscribe 106 views 3 years ago Learn how to use Pythagorean...

To calculate the distance from the start of a to the start of the lateral edge, all we need to do is find the hypotenuse of the right triangle. So: A^2 + B^2 = C^2. 1^2 + 2^2 = 5. so sqrt (5) is the distance between the start of A and the start of the lateral edge. So the base of our final triangle, b, is sqrt (5).

How to Solve Multi-Step Word Problems Involving the Pythagorean Theorem Step 1: Create a basic diagram for the context of the word problem, including a right triangle. Step 2: Label...

This video explains how to solve multi-step problems using pythagorean theorem.

8.2 Multi-step Pythagorean Theorem problems Name_____ ID: 1 Date_____ Period____ ©a c2A0n1v5M qKOuztAaN KSYoNf]tFwOa_rCeQ wLwLfCu.m y `AClGll jr\iJgLhvt^s_ HrJewsReDrXv_eTdF.-1-Find the area of each triangle. Round intermediate values to the nearest tenth. Use the rounded values to calculate the next value.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation [latex]\sqrt{m}[/latex] and defined it in this way: ... Step 1. Read the problem. Step 2. Identify what you are looking for. the length of the hypotenuse of the triangle: Step 3. Name.

Pythagorean theorem word problems. Google Classroom. You might need: Calculator. Steve is turning half of his backyard into a chicken pen. His backyard is a 24 meter by 45 meter rectangle. He wants to put a chicken wire fence that stretches diagonally from one corner to the opposite corner.

1. Define two points in the X-Y plane. The Pythagorean Theorem can easily be used to calculate the straight-line distance between two points in the X-Y plane. All you need to know are the x and y coordinates of any two points. Usually, these coordinates are written as ordered pairs in the form (x, y).

This calculator solves the Pythagorean Theorem equation for sides a or b, or the hypotenuse c. The hypotenuse is the side of the triangle opposite the right angle. For right triangles only, enter any two values to find the third. See the solution with steps using the Pythagorean Theorem formula. This calculator also finds the area A of the ...

Two cyclists start from the same location. One cyclist travels towards the north and the other towards the east, at the same speed. Find the speed of each in miles per hour if after two hours they ...

Step 1: Enter the values of any two angles and any one side of a triangle below for which you want to find the length of the remaining two sides. The Pythagorean theorem calculator finds the length of the remaining two sides of a given triangle using sine law or definitions of trigonometric functions.

Kuta Software - Infinite Geometry Name_____ Multi-Step Pythagorean Theorem Problems Date_____ Period____ Find the area of each triangle. Round intermediate values to the nearest tenth. Use the rounded values to calculate the next value. Round your final answer to the nearest tenth.

0:00 / 9:23 KutaSoftware: Geometry- Mulit-Step Pythagorean Theorem Problems Part 2 MaeMap 32.7K subscribers 8K views 5 years ago KutaSoftware: Geometry Worksheets Free worksheet at...

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