   Chapter 10.2, Problem 47E 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

Pentagon ABCDE has vertices A 0 ,   0 , B 0 ,   b , C d ,   c , D c ,   b , and E a ,   0 for positive numbers a , b , c , and d . Find an expression for the area of A B C D E .

To determine

To find:

An expression for the area of Pentagon ABCDE which has the vertices A0, 0, B0, b, Cd, c, Dc, b, and Ea, 0 for positive numbers a, b, c, and d.

Explanation

The pentagon ABCDE is formed with the vertices given as follows.

Mark the vertex A0, 0.

The vertex B0, b is marked in anywhere in the y-intercept.

The vertex Ea, 0 is marked anywhere in the x-intercept.

The vertex Cd, c is marked in the first quadrant, so that d<a and c>b.

Finally the vertex Dc, b is marked in the first quadrant between C and E, ensure that the points given in the vertex D are already marked by Cd, c and B0, b.

By joining these points, a pentagon is formed.

The area of pentagon can be calculated by finding the area of two trapezium and the area of triangle thus formed inside the pentagon.

This can be obtained by drawing the line from the vertex C which is perpendicular to x-axis and mark the x-intercept as G and draw the line from the vertex D which is also perpendicular to x-axis and mark the x-intercept as F.

The two trapezium are ABCG and GCDF.

The triangle thus formed is DFE.

Area of trapezium =base 1+base 2×height2

Base 1 and base 2 are the length of the two lines which are parallel.

Height is the distance between the two parallel lines.

Consider ABCG,

Base 1 AB is b.

Base 2 CG is c.

Height AG is d.

Area of trapezium ABCG=b+cd2

Consider GCDF,

Base 1 GC is c

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