   Chapter 10.4, Problem 17E 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.If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

To determine

The analytic proof for the given theorem “If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.”

Explanation

Given theorem is,

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

The above figure is drawn for the parallelogram ABCD with the diagonals perpendicular to each other.

Where, AC and BD are the diagonals of the parallelogram ABCD.

AO=CO

And BO=DO

where, O is the mid point of the two diagonals.

Considering the two triangles ADO and CDO,

A=C

AO=CO

So, the triangles ADO and CDO are congruent.

Based on the Side-Angle-Side rule, the hypotenuses of the triangles are equal in length.

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