   Chapter 8.5, Problem 9E 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

Suppose that a circle of radius r is inscribed in an equilateral triangle whose sides have length s. Find an expression for the area of the triangle in terms of r and s.(HINT: Use Theorem 8.5.3.)

To determine

To find:

An expression for the area of triangle in terms of r and s.

Explanation

Formula:

Area of a triangle with an inscribed circle:

If P is the perimeter of the triangle and r is the length of radius of its inscribed circle, then the area A of the triangle is given by

A=12rP

Calculation:

The radius of inscribed circle is r.

We know that, for a triangle perimeter is equal to the sum of lengths of all sides.

In an equilateral triangle, all the sides are of equal length.

The length of side of equilateral triangle is given as s.

So, perimeter of equilateral triangle P=s+s+s=3s

The formula of area A of a triangle with perimeter P and radius r of its inscribed circle is given by A=12rP

Let’s substitute the value of P in this formula to obtain an expression for area

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