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# The Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

The famous formula that resumes this phrase is c 2 = a 2 + b 2

Example 1

The measurements of three sides of a triangle are 5, 6, and 7 units. Determine whether the triangle is a right triangle.

Solution

Use the Pythagorean Theorem to see if the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

 c 2 = a 2 + b 2 7 2 5 2 + 6 2 The hypotenuse is the longest side. 49 25 + 36 49 61

Since 7 2 5 2 + 6 2 , the triangle is not a right triangle.

Sometimes, when working with the Pythagorean Theorem, one has to approximate square roots. Consider the following example.

Example 2

Find the length of the third side of the right triangle.

Solution To find the length of the third side, use the Pythagorean Theorem.

 c 2 = a 2 + b 2 Pythagorean Theorem 12 2 = a 2 + 8 2 Replace c with 12 and b with 8. 144 = a 2 + 64 144 - 64 = a 2 Subtract 64 from each side. 80 = a 2 = a Take the square root of each side. 8.9

The length of the leg is about 8.9 inches.

Try to find several problems involving the Pythagorean Theorem as it addresses geometric and algebraic concepts, and practices skills such as taking square roots.