how to do multi step pythagorean theorem problems

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 $

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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

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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 ‍  
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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

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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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