   Chapter 8.3, Problem 35E 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

Given a regular octagon R S T U V W X Y with each side of length 4 and diagonal R U ¯ , find the area of hexagon R Y X W V U .

To determine

To find:

The area of the hexagon RYXWVU.

Explanation

1) Area of the trapezium is

A=(a+b)2h,

where a, and b are the parallel sides of the trapezium and h is perpendicular distance between them.

2) The area of a rectangle is l×b, where l.is the length and b is the width of the rectangle.

Calculation:

Given,

A regular octagon RSTUVWXY with each side of length 4 and diagonal RU¯

Let O be the midpoint of the octagon.

From the figure UOV is the central angle of the regular octagon

therefore,

UOV=3608=45

Since UOV is a isosceles triangle,

mOVU=mOUV=180452=1352=67.5

Consider the right angle triangle RUV,

sinOVU=RURV

From the figure, RV=2OR

sinOVU=RU2ORRU=sinOVU2ORRU=sin67.52OR...(1)

Draw a line ON such that ON is Perpendicular to ST.

Since ONT is right angle triangle,

tanOTS=ONNT=ON42tanOTS=ON2ON=2tanOTS=2tan67.5=22

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