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# The Pythagorean Theorem

A 90Â° angle is a right angle, so a triangle that contains a 90Â° angle is called a right triangle. The side opposite the right angle is the hypotenuse of the triangle.

Let a and b represent the lengths of the sides that form the right angle. Let c represent the length of the hypotenuse.

The Pythagorean Theorem states the relationship between the lengths of the sides of a right triangle.

Formula â€” Pythagorean Theorem

If a and b are the lengths of the sides that form the right angle in a right triangle and c is the length of the hypoteneuse, then: c2 = a2 + b2

If we know the lengths of two sides of a right triangle, we can use the Pythagorean Theorem to find the length of the third side.

Example 1

Given a right triangle with sides of length 6 inches and 8 inches, find the length of the hypotenuse.

 Solution In the Pythagorean Theorem, substitute 6 inches for a and 8 inches for b. c2c2 = a2 + b2  = (6 inches)2 + (8 inches)2 Calculate the squares. Add. The value of c2 is 100. = 36 inches2 + 64 inches2 = 100 inches2 To find the value of c we ask, â€œWhat positive number squared is 100?â€ That is, â€œWhat is the square root of 100?â€ The square root of 100 is 10. c c = 10 inches

Therefore, the length of the hypotenuse, c, is 10 inches.

Example 2

Given a right triangle where one leg has length 3 units and the hypotenuse has length 5 units, find the length of the third side.

 Solution In the Pythagorean Theorem, substitute 3 for a and 5 for c. c2 52 = a2 + b2  = (3)2 + b2 Calculate the squares. Subtract 9 from both sides. 25 16 = 9 + b2= b2 We found that b2 is 16. The square root of 16 is 4. 4 = b
Therefore, the length of the third side, b, is 4 units.