Chapter 5.3, Problem 41E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Prove that the altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.

To determine

**To prove:**

The statement “The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.”

Explanation

**Definition:**

*AA:*

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

**Description:**

Draw an altitude in right triangle to the hypotenuse as shown below.

Figure

From the given figure, it is observed that altitude is drawn to the hypotenuse from the vertex which is perpendicular to the opposite side. Therefore, *ABD *and *DBC* are right triangles.

Here, the altitude in the original triangle *ABC* which separates two right triangles *ABD *and *DBC*.

It is enough to show that

First show that

Both triangles *ABC *and *DBC *are right triangles so *C.* That is,

The above mentioned *AA* definition,

Thus, It is proved that

Similarly show that

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