Chapter 8.3, Problem 35E

### 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 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)2â‹…h,

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 lÃ—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 RUÂ¯

Let O be the midpoint of the octagon.

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

therefore,

âˆ UOV=360âˆ˜8=45âˆ˜

Since â–³UOV is a isosceles triangle,

mâˆ OVU=mâˆ OUV=180âˆ˜âˆ’45âˆ˜2=135âˆ˜2=67.5âˆ˜

Consider the right angle triangle â–³RUV,

sinâˆ OVU=RURV

From the figure, RV=2â‹…OR

sinâˆ OVU=RU2â‹…ORRU=sinâˆ OVUâ‹…2ORRU=sin67.5âˆ˜â‹…2ORâ€‰â€‰â€‰...(1)

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

Since â–³ONT is right angle triangle,

tanâ€‰âˆ OTS=ONNT=ON42tanâ€‰âˆ OTS=ON2ON=2â‹…tanâ€‰âˆ OTS=2â‹…tanâ€‰67.5âˆ˜=2â‹…2

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