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Pythagorean Theorem and Pythagorean Triples

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What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one? After completing this Concept, you'll be able to use the Pythagorean Theorem to solve problems like this one.

Watch This

CK-12 Foundation: Using The Pythagorean Theorem

James Sousa: Pythagorean Theorem

Guidance

The two shorter sides of a right triangle (the sides that form the right angle) are the legs and the longer side (the side opposite the right angle) is the hypotenuse . For the Pythagorean Theorem, the legs are “ a ” and “ b ” and the hypotenuse is “ c ”.

Pythagorean Theorem: Given a right triangle with legs of lengths a and b and a hypotenuse of length c , a^2+b^2=c^2 .

For proofs of the Pythagorean Theorem go to: http://www.mathsisfun.com/pythagoras.html and scroll down to “And You Can Prove the Theorem Yourself.”

The converse of the Pythagorean Theorem is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.

Pythagorean Theorem Converse: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If a^2 + b^2 = c^2 , then \triangle ABC is a right triangle.

Pythagorean Triples

A combination of three numbers that makes the Pythagorean Theorem true is called a Pythagorean triple. Each set of numbers below is a Pythagorean triple.

3, 4, 5 && 5, 12, 13	&& 7, 24, 25 && 8, 15, 17 && 9, 12, 15 && 10, 24, 26

Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.

Example A

Do 6, 7, and 8 make the sides of a right triangle?

Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, c . If 6^2+7^2=8^2 , then they are the sides of a right triangle.

6^2 + 7^2 &= 36 + 49=85\\8^2 &= 64 && 85 \neq 64, \ \text{so the lengths are not the sides of a right triangle.}

Example B

Find the length of the hypotenuse.

Use the Pythagorean Theorem. Set a = 8 and b  = 15 . Solve for c .

8^2 + 15^2 &= c^2\\64 + 225 &= c^2\\289 &= c^2 && Take \ the \ square \ root \ of \ both \ sides.\\17 &= c

Example C

Is 20, 21, 29 a Pythagorean triple?

If 20^2 + 21^2 = 29^2 , then the set is a Pythagorean triple.

20^2 + 21^2 &= 400+441=841\\29^2 &= 841

Therefore, 20, 21, and 29 is a Pythagorean triple.

Example D

Determine if the triangles below are right triangles.

a)

b)

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent c .

a) a^2 + b^2 & = c^2\\8^2 + 16^2 & \overset{?} = \left( 8 \sqrt{5} \right )^2\\64 + 256 & \overset{?} = 64 \cdot 5\\320 &= 320 \qquad \text{Yes}

b) a^2+b^2 & = c^2\\22^2 + 24^2 & \overset{?}= 26^2\\484 + 576 & \overset{?}= 676\\1060 & \neq 676 \qquad \text{No}

CK-12 Foundation: Using The Pythagorean Theorem

Guided Practice

1. Find the missing side of the right triangle below.

2. What is the diagonal of a rectangle with sides 10 and 16?

3. Do the following lengths make a right triangle?

a) \sqrt{5}, 3, \sqrt{14}

b) 6, 2 \sqrt{3}, 8

c) 3 \sqrt{2}, 4 \sqrt{2}, 5\sqrt{2}

Answers:

1. Here, we are given the hypotenuse and a leg. Let’s solve for b .

7^2 + b^2 &= 14 ^2\\49 + b^2 &= 196\\b^2 &= 147\\b &= \sqrt{147} = \sqrt{49 \cdot 3} = 7 \sqrt{3}

2. For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find d .

10^2 + 16^2 &= d^2\\100 + 256 &= d^2\\356 &= d^2\\d &= \sqrt{356} = 2 \sqrt{89} \approx 18.87

3. Even though there is no picture, you can still use the Pythagorean Theorem. Again, the longest length will be c .

a) & \left( \sqrt{5} \right )^2 + 3^2 = \sqrt{14}^2\\& 5+9 = 14\\& \text{Yes}

b) 6^2 + \left( 2 \sqrt{3} \right )^2 & = 8^2\\36+(4 \cdot 3) & = 64\\36+12 & \neq 64

c) This is a multiple of \sqrt{2} of a 3, 4, 5 right triangle. Yes, this is a right triangle.

Practice

Find the length of the missing side. Simplify all radicals.

  1. If the legs of a right triangle are 10 and 24, then the hypotenuse is __________.
  2. If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
  3. If the sides of a square are 16, then the diagonal is ____________.
  4. If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

  1. 12, 35, 37
  2. 9, 17, 18
  3. 10, 15, 21
  4. 11, 60, 61
  5. 15, 20, 25
  6. 18, 73, 75

Determine if the following lengths make a right triangle.

  1. 7, 24, 25
  2. \sqrt{5}, 2 \sqrt{10}, 3 \sqrt{5}
  3. 2 \sqrt{3}, \sqrt{6}, 8
  4. 15, 20, 25
  5. 20, 25, 30
  6. 8 \sqrt{3}, 6, 2 \sqrt{39}

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