Chapter 4.1, Problem 46E

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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,

In âˆ†ADE,

In âˆ†BCF,

Since,

Since

In âˆ†ACF,

In âˆ†BDE,

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