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Solving with the Pythagorean Theorem

5 min readdecember 13, 2021

Solving Right Triangles with the Pythagorean Theorem

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0jgqeT8cmx2c.jpg?alt=media&token=9c59e108-85da-4b11-92dd-987f1bedbcd8

(Image courtesy of Pixabay

What is the Pythagorean theorem?

Right triangles are everywhere, and one example of the Pythagorean theorem in use in the real world is in sports. If you have ever watched a baseball game, and have witnessed the ball being thrown from home base to second base, then you have witnessed a formation of a right triangle! With this right triangle formed, you can actually calculate the distance the baseball traveled from home base to second base given the distances between the bases (the side lengths of the right triangle).  

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0pgEgDxdKfRs.png?alt=media&token=d071e651-249e-41c8-84b6-a6cb7cb63b3d

Mathematicians who have long studied geometry have found an equation to calculate every side length of a triangle easily. With the Pythagorean theorem, we can easily find the length of each side of a right triangle. 

What You Need to Remember:

The Pythagorean theorem applies to ALL right triangles.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-kSsKRxvxQW6p.gif?alt=media&token=0491227d-c4f6-4d20-94f3-60fb43a1af12

(GIF courtesy of Giphy)

The Pythagorean theorem is an equation used in math that can help find the hypotenuse, the side that sits opposite the triangle's right angle. Also, it is important to note that the hypotenuse is the longest side of the right triangle. The hypotenuse can be found by squaring both sides of a right triangle and equating it to the hypotenuse squared. To find the length of either side of a right triangle, you can arrange the equation as:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-aCb1hsqwIdl5.png?alt=media&token=c82f2f7d-1e16-42c0-9597-ce3145baf3ae

where c is the hypotenuse, and a and b are the adjacent and opposite angles (it does not matter which is which, as long as c is the hypotenuse) 

You may have seen this equation in your math class before. We can also use the Pythagorean theorem to find the length of the opposite or adjacent angles. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-f0YYZfdUO6js.png?alt=media&token=545c5774-a5c6-46ad-98d5-832ed9b0835d

Finding the length of c

The length of side c is the square root of a squared plus b squared:

c=√(a^2+b^2)

What You Need to Remember:

Side c is always the hypotenuse of the right triangle!

Finding the length of a

The length of side a is the square root of the hypotenuse squared minus b squared:

a=√(c^2-b^2)

Finding the length of b

The length of side b is equal to the square root of c squared minus a squared:

b=√(c^2-a^2)

As shown above, the Pythagorean theorem equation can be manipulated to find the side length that you want. For example, to find the side length of a, the equation is rearranged by subtracting b^2 from the left side, then taking the square root of the right side of the equation. This gives us a clean equation to solve to find the side length of a. This is the same method we use to find b. 

Now that you have seen how to find each side length, let's test your skills! Complete the practice problem below to test your proficiency:

If a right triangle has an adjacent side length of 4 inches and an opposite side length of 3 inches, what is the length of the hypotenuse?

  • Hint: Use the Pythagorean formula to help you find the hypotenuse length!

  • c=√(4^2+3^2)

  • c=√(16 +9)

  • c=√25

  • c=5

Were you able to answer this correctly? If not, try practicing more problems and remember to use the Pythagorean equation for reference. 

Right Triangles

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-61KoJtXOuD7I.png?alt=media&token=2ef84870-f400-462b-8a99-dc677f0fbb13

Now that we have covered the basics of the Pythagorean theorem, it is essential to understand the types of triangles that can be solved.

A right triangle is a type of triangle in which one of the interior angles is 90 degrees. A right angle is a 90-degree angle that is represented by a small box in the interior of an angle. As shown above, angle C is a right angle. 

 

Right triangles belong to two categories:

  • Isosceles right triangle: In an isosceles triangle, one interior angle is 90 degrees while the other two interior angles are each 45 degrees. An easy way to remember this is by thinking of “45-90-45” in terms of the degrees of the interior angles. 

  • Scalene right triangle: In a scalene triangle, all of its sides are different lengths. In other words, all the sides of the scalene triangle are unequal, and all of the angles also unequal. However, a scalene right triangle must have one 90-degree angle.

    https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-b9YtJhiNFHOM.png?alt=media&token=5b50b071-e485-4093-9f6c-9836021cb731

Opposite Side

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-wg09uWWg42Xs.png?alt=media&token=5daa37e1-04da-4366-a10e-2fa4ae76f0ea

The opposite side is the length that is directly across from the specified angle, which is often marked as theta. When using the Pythagorean theorem, It does not matter if a or b is assigned to the opposite side; the only requirement is that c is the hypotenuse! When trying to differentiate between opposite, adjacent, or hypotenuse, remember to reference the angle the problem is asking for, not necessarily the right angle.

Adjacent Side

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4wKEfe7Wf1d6.png?alt=media&token=bee36e15-73af-45ac-8a44-73a583ccebbd

The adjacent side is the side that is next to the specified angle. 

Hypotenuse

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-YHEDErFQUN6Z.png?alt=media&token=4918fd31-999f-4919-86c1-8dd8d237ff06

The hypotenuse side is the side across from the 90 degree angle. Remember that it is always the longest side of the triangle!

Importance of the Pythagorean Theorem

The Pythagorean theorem has many implications in everyday life and higher level math. Architects and engineers use the Pythagorean theorem on everyday projects to evaluate the lengths of complex shapes. The Pythagorean theorem is also useful for construction—many bridges, for example, are made up of triangles. Because of their shape, they evenly distribute force that is applied, increasing safety and stability.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-5dVKH2IddiA9.jpg?alt=media&token=8a08eb8b-c119-43b6-8778-09b0197cbe54

(Image courtesy of Pixabay)

This theorem can also be helpful for solving navigation word problems. For example, if you are at sea and travel 500 miles north and 700 miles west, you can use the Pythagorean theorem to find how far your ship is from your starting point by calculating the hypotenuse. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-kf99DigxVWBp.gif?alt=media&token=125cabf5-fa7a-4d30-aa99-4980655ec5c9

(GIF courtesy of Giphy)

Opposite, adjacent, and hypotenuse side lengths are also crucial for trigonometry, in which they appear again when solving sine, cosine, and tangent equations. 

🤝Connect with other students studying Geometry with Hours

Solving with the Pythagorean Theorem

5 min readdecember 13, 2021

Solving Right Triangles with the Pythagorean Theorem

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0jgqeT8cmx2c.jpg?alt=media&token=9c59e108-85da-4b11-92dd-987f1bedbcd8

(Image courtesy of Pixabay

What is the Pythagorean theorem?

Right triangles are everywhere, and one example of the Pythagorean theorem in use in the real world is in sports. If you have ever watched a baseball game, and have witnessed the ball being thrown from home base to second base, then you have witnessed a formation of a right triangle! With this right triangle formed, you can actually calculate the distance the baseball traveled from home base to second base given the distances between the bases (the side lengths of the right triangle).  

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0pgEgDxdKfRs.png?alt=media&token=d071e651-249e-41c8-84b6-a6cb7cb63b3d

Mathematicians who have long studied geometry have found an equation to calculate every side length of a triangle easily. With the Pythagorean theorem, we can easily find the length of each side of a right triangle. 

What You Need to Remember:

The Pythagorean theorem applies to ALL right triangles.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-kSsKRxvxQW6p.gif?alt=media&token=0491227d-c4f6-4d20-94f3-60fb43a1af12

(GIF courtesy of Giphy)

The Pythagorean theorem is an equation used in math that can help find the hypotenuse, the side that sits opposite the triangle's right angle. Also, it is important to note that the hypotenuse is the longest side of the right triangle. The hypotenuse can be found by squaring both sides of a right triangle and equating it to the hypotenuse squared. To find the length of either side of a right triangle, you can arrange the equation as:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-aCb1hsqwIdl5.png?alt=media&token=c82f2f7d-1e16-42c0-9597-ce3145baf3ae

where c is the hypotenuse, and a and b are the adjacent and opposite angles (it does not matter which is which, as long as c is the hypotenuse) 

You may have seen this equation in your math class before. We can also use the Pythagorean theorem to find the length of the opposite or adjacent angles. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-f0YYZfdUO6js.png?alt=media&token=545c5774-a5c6-46ad-98d5-832ed9b0835d

Finding the length of c

The length of side c is the square root of a squared plus b squared:

c=√(a^2+b^2)

What You Need to Remember:

Side c is always the hypotenuse of the right triangle!

Finding the length of a

The length of side a is the square root of the hypotenuse squared minus b squared:

a=√(c^2-b^2)

Finding the length of b

The length of side b is equal to the square root of c squared minus a squared:

b=√(c^2-a^2)

As shown above, the Pythagorean theorem equation can be manipulated to find the side length that you want. For example, to find the side length of a, the equation is rearranged by subtracting b^2 from the left side, then taking the square root of the right side of the equation. This gives us a clean equation to solve to find the side length of a. This is the same method we use to find b. 

Now that you have seen how to find each side length, let's test your skills! Complete the practice problem below to test your proficiency:

If a right triangle has an adjacent side length of 4 inches and an opposite side length of 3 inches, what is the length of the hypotenuse?

  • Hint: Use the Pythagorean formula to help you find the hypotenuse length!

  • c=√(4^2+3^2)

  • c=√(16 +9)

  • c=√25

  • c=5

Were you able to answer this correctly? If not, try practicing more problems and remember to use the Pythagorean equation for reference. 

Right Triangles

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-61KoJtXOuD7I.png?alt=media&token=2ef84870-f400-462b-8a99-dc677f0fbb13

Now that we have covered the basics of the Pythagorean theorem, it is essential to understand the types of triangles that can be solved.

A right triangle is a type of triangle in which one of the interior angles is 90 degrees. A right angle is a 90-degree angle that is represented by a small box in the interior of an angle. As shown above, angle C is a right angle. 

 

Right triangles belong to two categories:

  • Isosceles right triangle: In an isosceles triangle, one interior angle is 90 degrees while the other two interior angles are each 45 degrees. An easy way to remember this is by thinking of “45-90-45” in terms of the degrees of the interior angles. 

  • Scalene right triangle: In a scalene triangle, all of its sides are different lengths. In other words, all the sides of the scalene triangle are unequal, and all of the angles also unequal. However, a scalene right triangle must have one 90-degree angle.

    https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-b9YtJhiNFHOM.png?alt=media&token=5b50b071-e485-4093-9f6c-9836021cb731

Opposite Side

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-wg09uWWg42Xs.png?alt=media&token=5daa37e1-04da-4366-a10e-2fa4ae76f0ea

The opposite side is the length that is directly across from the specified angle, which is often marked as theta. When using the Pythagorean theorem, It does not matter if a or b is assigned to the opposite side; the only requirement is that c is the hypotenuse! When trying to differentiate between opposite, adjacent, or hypotenuse, remember to reference the angle the problem is asking for, not necessarily the right angle.

Adjacent Side

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4wKEfe7Wf1d6.png?alt=media&token=bee36e15-73af-45ac-8a44-73a583ccebbd

The adjacent side is the side that is next to the specified angle. 

Hypotenuse

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-YHEDErFQUN6Z.png?alt=media&token=4918fd31-999f-4919-86c1-8dd8d237ff06

The hypotenuse side is the side across from the 90 degree angle. Remember that it is always the longest side of the triangle!

Importance of the Pythagorean Theorem

The Pythagorean theorem has many implications in everyday life and higher level math. Architects and engineers use the Pythagorean theorem on everyday projects to evaluate the lengths of complex shapes. The Pythagorean theorem is also useful for construction—many bridges, for example, are made up of triangles. Because of their shape, they evenly distribute force that is applied, increasing safety and stability.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-5dVKH2IddiA9.jpg?alt=media&token=8a08eb8b-c119-43b6-8778-09b0197cbe54

(Image courtesy of Pixabay)

This theorem can also be helpful for solving navigation word problems. For example, if you are at sea and travel 500 miles north and 700 miles west, you can use the Pythagorean theorem to find how far your ship is from your starting point by calculating the hypotenuse. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-kf99DigxVWBp.gif?alt=media&token=125cabf5-fa7a-4d30-aa99-4980655ec5c9

(GIF courtesy of Giphy)

Opposite, adjacent, and hypotenuse side lengths are also crucial for trigonometry, in which they appear again when solving sine, cosine, and tangent equations. 

🤝Connect with other students studying Geometry with Hours



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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.