   Chapter 8.5, Problem 10E 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 a circle of radius r is inscribed in a rhombus, each of whose sides has length s. Find an expression for the area of the rhombus in terms of r and s.

To determine

To find:

An expression for area of rhombus

Explanation

Formula:

Area of triangle:

If b and h are lengths of base and height of triangle respectively, then area A of triangle is given by the formula,

A=12bh.

Properties of rhombus:

Rhombus is a parallelogram with all equal sides.

The diagonals of a rhombus are perpendicular bisectors of each other.

Calculation:

Consider the following diagram in which ABCD is a rhombus with an inscribed circle.

It is given that length of side of rhombus is s. We know that all sides of a rhombus are equal. Hence, in the above diagram, AB=BC=CD=AD=s.

Let’s find the area of AOD.

For finding the area, let’s consider AD as the base of triangle. Thus, base AD=s.

If AD is the base, OE is the corresponding height. It is clear from the diagram that OE is the radius of circle. The radius of circle is given as r. Thus, height of AOD=r

Area of AOD=12bh

Area of AOD=12×s×r=sr2

Consider AOD and AOB

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