   Chapter 8.5, Problem 23E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

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

In Exercises 23 and 24, find the exact areas of the shaded regions. To determine

To find:

The exact area of the shaded region.

Explanation

Formula:

Area of sector:

If r is the radius of the circle, the area A of a sector whose arc has degree measure m is given by

A=m360×πr2

Heron’s formula for area of triangle:

If a, b and c are lengths of sides of triangle, then area of triangle is given by the formula:

A=s(s-a)(s-b)(s-c)

Where s is the semi perimeter which is given by s=12(a+b+c)

Calculation:

To find the area of the shaded region, we have to subtract the area of 3 sectors from the area of triangle.

Let’s find the area of triangle.

The given triangle is an equilateral triangle. Hence, a=b=c=10 cm.

Semi perimeter of the triangle is s=10+10+102=302=15 cm.

So, let’s substitute these values in the formula to find the area of triangle.

Atriangle=s(s-a)(s-b)(s-c)

Atriangle=1515-1015-1015-10

Atriangle=15×5×5×5

Atriangle=253 cm2.

The arcs are drawn in such a way that they intersect the sides of triangle at their midpoints

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