   Chapter 10.4, Problem 2E 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

In Exercises 1 to 17, complete an analytic proof for each theorem.The opposite sides of a parallelogram are equal in length.

To determine

The analytic proof for the given theorem “The opposite sides of a parallelogram are equal in length”.

Explanation

Given theorem is,

The opposite sides of a parallelogram are equal in length.

The above graph shows the parallelogram ABCD.

From the theorem the length of the opposite sides of the parallelogram are equal and expressed as below,

Opposite side of AB is CD.

And the opposite side of AD is BC.

The coordinates of the figure are A0, 0, Ba,0, Ca+b, c and D(b, c).

In order to prove “The opposite sides of a parallelogram are equal in length”, needed to use distance formula.

Length of AB= length of CD.

Using distance formula for finding length of the diagonal AC.

Where, A0, 0 and Ba,0

AB=(a-0)2+(0-0)2

Simplifying the above equation,

AB=(a)2

AB=a

Similarly, Ca+b, c and D(b, c)

CD=b-a+b2+(c-c)2

Simplifying the above equation,

CD=(b-a-b)2+(0)2

CD=(-a)2=a2

CD=a

Hence, AB=CD is proved

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