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Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
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The Pythagorean Theorem
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The Pythagorean Theorem

A 90° angle is a right angle, so a triangle that contains a 90° 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 — 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.


In the Pythagorean Theorem, substitute 6 inches for a and 8 inches for b.



 = a2 + b2

 = (6 inches)2 + (8 inches)2

Calculate the squares.


The value of c2 is 100.

   = 36 inches2 + 64 inches2

= 100 inches2

To find the value of c we ask,

“What positive number squared is 100?” That is, “What is the square root of 100?”

The square root of 100 is 10.



= 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.


In the Pythagorean Theorem, substitute 3 for a and 5 for c.



 = a2 + b2

 = (3)2 + b2

Calculate the squares.

Subtract 9 from both sides.



 = 9 + b2

= b2

We found that b2 is 16.

The square root of 16 is 4.




= b

Therefore, the length of the third side, b, is 4 units.
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