   Chapter 10.2, Problem 37E 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

If 2 ,   3 , 5 ,   - 2 , and 7 ,   2 are three vertices (not necessarily consecutive) of a parallelogram, find the possible locations of the fourth vertex.

To determine

To find:

The possible location of the fourth vertex for the given three vertices 2, 3, 5, -2, and 7, 2 of a parallelogram.

Explanation

Let the three vertices of a parallelogram are given as A2, 3, B5, -2, and C7, 2.

There may be three possibilities of the fourth vertex.

AB be the side and AC be the diagonal.

BC be the side and AB be the diagonal.

AB be the side and BC be the diagonal.

As this is a parallelogram, the midpoint of diagonals will be equal.

Consider the first possibility,

The midpoint of AC is

The midpoint of the diagonal AC that contains A2, 3, and C7, 2 is

x, y=x1+x22, y1+y22

x, y=2+72, 3+22

x, y=92, 52

x, y=92, 52

The midpoint of diagonal BD is same as AC.

92, 52=5+x22, -2+y22

Thus,

Equate the first term

5+x22=92

5+x2=9

x2=9-5

x2=4

Equate the second term

-2+y22=52

-2+y2=5

y2=5+2

y2=7

The first possible vertex is 4, 7.

Consider the second possibility,

The midpoint of the diagonal AB that contains A2, 3, and B5, -2 is

x, y=x1+x22, y1+y22

x, y=2+52, 3-22

x, y=72, 12

The midpoint of diagonal CD is same as AB

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