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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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BuyFindarrow_forward

Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 23 and 24, find the exact areas of the shaded regions.

Chapter 8.5, Problem 23E, 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.

Ashaded region=Atriangle-Athree sectors

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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