   Chapter 10.CR, Problem 25CR 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

Three of the four vertices of a parallelogram are 0 ,   - 2 , 6 ,   8 , and 10 ,   1 . Find the possibilities for the coordinates of the remaining vertex.

To determine

To find:

The possibilities of the fourth coordinates of a parallelogram with the three vertices are 0, -2, 6, 8, and 10, 1.

Explanation

Let the three vertices of a parallelogram are given as A0, -2, B6, 8, and C10, 1.

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 the diagonal AC that contains A0, -2, and C10, 1 is

x, y=x1+x22, y1+y22

x, y=0+102, -2+12

x, y=102, -12

x, y=5, -12

The midpoint of diagonal BD is same as AC.

5, -12=6+x22, 8+y22

Equate the first term

6+x22=5

6+x2=10

x2=10-6

x2=4

Equate the second term

8+y22=-12

8+y2=-1

y2=-1-8

y2=-9

The first possible vertex is 4, -9.

Consider the second possibility,

The midpoint of the diagonal AB that contains A0, -2, and B6, 8 is

x, y=x1+x22, y1+y22

x, y=0+62, -2+82

x, y=62, 62

x, y=3, 3

The midpoint of diagonal CD is same as AB

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