Chapter 8.3, Problem 37E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

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

# Given regular pentagon R S T V Q and equilateral triangle P Q R , and the length of an apothem (not shown) of R S T V Q is 12 , while the length of each side of the equilateral triangle is 10 , find the approximate area of the kite V P S T .

To determine

To find:

The approximate area of the kite VPST.

Explanation

1) The perimeter of a regular polygon is given by P = ns, where n is the number of sides and s is the length of any side.

2) The area of a regular polygon with apothem a and perimeter P is given by A=12(aP).

3) Area of the triangle using sine function is,

A=abâ€‰sinC2

where a, and b are the lengths of two sides of the triangle, C is the included angle.

Calculation:

Given,

A regular pentagon RSTVQ

and the length of an apothem is 12, and the length of each side of the equilateral triangle is 10

Area of the regular pentagon RSTVQ is A=12(aP)

Substitute a=12 and P=5Ã—10=50 in A=12(aP)

ARSTVQ=12(12â‹…50)=6â‹…50ARSTVQ=300

Since RSTVQ is a regular pentagon,

mâˆ VQR=180âˆ˜(5âˆ’2)5=180âˆ˜(3)5=108âˆ˜

From the figure, mâˆ VQP=mâˆ VQRâˆ’mâˆ PQR

given PQR is an equilateral triangle

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