Chapter 5.3, Problem 42E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Prove that the line segment joining the midpoints of two sides of a triangle determines a triangle that is similar to the original triangle.

To determine

**To prove:**

The statement “The line segment joining the midpoints of two sides of a triangle determines a triangle that is similar to the original triangle.”

Explanation

**Definition:**

If the three sides of one triangle are proportional (in length) to the three corresponding sides of a second triangle, then the triangle are similar.

**Description:**

Draw the line segment joining the midpoints of two sides as shown below.

Figure

From the given figure, it is observed that the line segment *DE* joining the two sides of *AB *and *AC*. In the triangle *ABC, D *is the midpoint of *AB *and *E *is the midpoint of *AC*.

By the definition of midpoint,

It is known that the line segment *DE* joins the midpoints of two sides a triangle, its length *DE *is half the length of *BC*. That is,

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