   Chapter 4.1, Problem 46E 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

Prove: In a parallelogram, the sum of the squares of the lengths of its diagonals is equal to the sum of the squares of the lengths of its sides.

To determine

To prove:

In a parallelogram, the sum of the squares of the lengths of its diagonals is equal to the sum of the squares of the lengths of its sides.

Explanation

Given:

A parallelogram with diagonals.

Definition:

1. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

2. An altitude of a parallelogram is any line segment drawn from one side so that it is perpendicular to the nonadjacent sides or to an extension of that side.

Approach:

Let ABCD be a parallelogram with the diagonals AC and BD.

Draw altitudes DE to meet AB at E and CF to meet AB produced at F.

Since the altitudes are anywhere equal, DE=CF.

In ∆BCF, BC2=BF2+CF2

AE2+DE2=BF2+CF2

Since DE=CF

AE2=BF2

AE=BF

In ∆ACF, AC2=AF2+CF2

AC2=(AB+BF)2+BC2-BF2

AC2=AB2+BF2+BC2-BF2

AC2=AB2+BF2+2AB·BF+BC2-BF2

AC2=AB2+2AB·BF+BC2…(1)

In ∆BDE, BD2=BE2+

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