Chapter 4.3, Problem 37E

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6th Edition

Daniel C. Alexander + 1 other

ISBN: 9781285195698

Textbook Problem

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*In Exercises 32 to 37, write a formal proof of each theorem.*

If the midpoints of the sides of a rectangle are joined in order, the quadrilateral formed is a rhombus.

To determine

**To write:**

A formal proof of the given theorem.

Explanation

**Calculation:**

Given: If the midpoints of the sides of a rectangle are joined in order, the quadrilateral formed is a rhombus.

Consider the rectangle *ABCD* and the quadrilateral formed by joining the midpoints *E, F, G, H* of the sides of the rectangle.

Consider a rectangle with its longest side being horizontal.

The rhombus will form four triangles, each in a corner of the rectangle.

By the definition of midpoint, the two sides next to the 90 degree angles for the all of the triangles will be congruent.

By SAS, this means that the triangles are congruent.

Since sides opposite to each other are congruent, it is a parallelogram.

Since adjacent sides are congruent and it is a parallelogram, it is a rhombus.

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