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Pythagorean Theorem and Pythagorean Triples

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What if a friend of yours wanted to design a rectangular building with one wall 65 ft long and the other wall 72 ft long? How can he ensure the walls are going to be perpendicular? After completing this Concept, you'll be able to apply the Pythagorean Theorem in order to solve problems like these.

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CK-12 Foundation: Chapter8ThePythagoreanTheoremandPythagoreanTriplesA

James Sousa: Pythagorean Theorem

Guidance

The sides of a right triangle are called legs (the sides of the right angle) and 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 , then a^2 + b^2 = c^2 .

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.

There are several proofs of the Pythagorean Theorem, shown below.

Investigation: Proof of the Pythagorean Theorem

Tools Needed: pencil, 2 pieces of graph paper, ruler, scissors, colored pencils (optional)

  1. On the graph paper, draw a 3 in. square, a 4 in. square, a 5 in square and a right triangle with legs of 3 and 4 inches.
  2. Cut out the triangle and square and arrange them like the picture on the right.
  3. This theorem relies on area. Recall from a previous math class, that the area of a square is length times width. But, because the sides are the same you can rewrite this formula as A_{square} = length \times width = side \times side = side^2 . So, the Pythagorean Theorem can be interpreted as (square \ with \ side \ a)^2 + (square \ with \ side \ b)^2 = (square \ with \ side  \ c)^2 . In this Investigation, the sides are 3, 4 and 5 inches. What is the area of each square?
  4. Now, we know that 9 + 16 = 25 , or 3^2 + 4^2 = 5^2 . Cut the smaller squares to fit into the larger square, thus proving the areas are equal.
Another Proof of the Pythagorean Theorem

This proof is “more formal,” meaning that we will use letters, a, b, and c to represent the sides of the right triangle. In this particular proof, we will take four right triangles, with legs a and b and hypotenuse c and make the areas equal.

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

Pythagorean Triples

A Pythagorean Triple is a set of three whole numbers that makes the Pythagorean Theorem true. The most frequently used Pythagorean triple is 3, 4, 5, as in Investigation 8-1. Any multiple of a Pythagorean triple is also considered a triple because it would still be three whole numbers. Therefore, 6, 8, 10 and 9, 12, 15 are also sides of a right triangle. Other Pythagorean triples are:

3, 4, 5 \qquad 5, 12, 13  \qquad 7, 24, 25 \qquad 8, 15, 17

There are infinitely many Pythagorean triples. To see if a set of numbers makes a triple, plug them into the Pythagorean Theorem.

Example A

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

Plug in the three numbers into the Pythagorean Theorem. The largest length will always be the hypotenuse . 6^2 + 7^2 = 36 + 49 = 85 \neq 8^2 . Therefore, these lengths do not make up the sides of a right triangle.

Example B

Find the length of the hypotenuse of the triangle below.

Let’s use the Pythagorean Theorem. Set a and b equal to 8 and 15 and solve for c , the hypotenuse.

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

When you take the square root of an equation, usually the answer is +17 or -17. Because we are looking for length, we only use the positive answer. Length is never negative .

Example C

Is 20, 21, 29 a Pythagorean triple?

If 20^2 + 21^2 is equal to 29^2 , then the set is a 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 triangle below is a right triangle.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent c , in the equation.

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

The triangle is a right triangle.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter8PythagoreanTheoremandPythagoreanTriplesB

Concept Problem Revisited

To make the walls perpendicular, find the length of the diagonal.

65^2 + 72^2 & = c^2\\4225 + 5184 & = c^2\\9409 & = c^2\\97 & = c

In order to make the building rectangular, both diagonals must be 97 feet.

Vocabulary

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 . The Pythagorean Theorem states that a^2+b^2=c^2 , where the legs are “ a ” and “ b ” and the hypotenuse is “ c ”. A combination of three numbers that makes the Pythagorean Theorem true is called a Pythagorean triple.

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 \sqrt{5} ?

3. Determine if the triangle below is a right triangle.

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{7 \cdot 7 \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 + \left ( 16 \sqrt{5} \right )^2 & = d^2\\100 + 1280 & = d^2\\1380 & = d^2\\d & = \sqrt{1380} = 2 \sqrt{345}

3. Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent c , in the equation.

a^2 + b^2 &= c^2\\22^2 + 24^2&= 26^2\\484 + 576 &= 676\\1060 &\neq 676

The triangle is not a right triangle.

Explore More

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 legs of a right triangle are x and y , then the hypotenuse 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

Pythagorean Theorem Proofs

The first proof below is similar to the one done earlier in this Concept. Use the picture below to answer the following questions.

  1. Find the area of the square with sides (a + b) .
  2. Find the sum of the areas of the square with sides c and the right triangles with legs a and b .
  3. The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.

Major General James A. Garfield (and former President of the U.S) is credited with deriving this next proof of the Pythagorean Theorem using a trapezoid.

  1. Find the area of the trapezoid using the trapezoid area formula: A = \frac{1}{2} (b_1 + b_2)h
  2. Find the sum of the areas of the three right triangles in the diagram.
  3. The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.

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