Chapter 5.3, Problem 41E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

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, BÂ¯âŠ¥ACÂ¯ and the two triangles 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 Î”ABCâˆ¼Î”DBCâˆ¼Î”ABD.

First show that Î”ABCâˆ¼Î”DBC.

Both triangles ABC and DBC are right triangles so âˆ Bâ‰…âˆ D and both triangles have the common point C. That is, âˆ Câ‰…âˆ C by identity.

The above mentioned AA definition, Î”ABCâˆ¼Î”DBC since two angles of one triangle are congruent to two angles of another triangle.

Thus, It is proved that Î”ABCâˆ¼Î”DBC.

Similarly show that Î”ABCâˆ¼Î”ABD

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