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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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Section
BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Prove that if semicircles are constructed on each of the sides of a right triangle, then the area of the semicircle on the hypotenuse is equal to the sum of the areas of the semicircles on the two legs.

To determine

To prove:

If semicircles are constructed on each of the sides of a right triangle, then the area of the semicircle on the hypotenuse is equal to the sum of the areas of the semicircles on the two legs.

Explanation

Formula:

The area of the semi-circle A=12πr2

Pythagorean Theorem:

Regardless of what figure we construct on the sides of the right triangle, as long as the figures are similar, the sum of the areas constructed on the shorter sides (the legs) would equal the area of the square on the longer side (the hypotenuse) then we will always get back to a2+b2=c2.

Calculation:

Consider the following diagram:

To find the area of the semi-circle with diameter b:

Radius r=b2

Area of the semi-circle A=12πr2

A=12π(b2)2=12π×b24=πb28

The area of the semi-circle with diameter b = πb28

To find the area of the semi-circle with diameter a:

A=12π(a2)2=12π×a24=πa28

To find the area of the semi-circle with diameter c:

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