Question

use the figure shown to the right to prove the following. given: right \\( \triangle a b c \\) with altitude to the hypotenuse \\( \overline{c d} \\) prove: \\( \frac{a b}{a c}=\frac{a c}{a d}, \frac{a b}{b c}=\frac{b c}{d b} \\) statements 1. right \\( \triangle a b c \\) with altitude to the hypotenuse \\( \overline{c d} \\). 2. 1. given 2. altitude of a right triangle

Explanation:

Step1: Prove \(\triangle ABC\sim\triangle ACD\)

In right \(\triangle ABC\) and right \(\triangle ACD\), \(\angle A=\angle A\) (common angle). By AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle ACD\).
Since the triangles are similar, the ratios of their corresponding sides are equal. So, \(\frac{AB}{AC}=\frac{AC}{AD}\).

Step2: Prove \(\triangle ABC\sim\triangle CBD\)

In right \(\triangle ABC\) and right \(\triangle CBD\), \(\angle B=\angle B\) (common angle). By AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle CBD\).
Since the triangles are similar, the ratios of their corresponding sides are equal. So, \(\frac{AB}{BC}=\frac{BC}{DB}\).

Answer:

The proof is completed by using the AA similarity criterion for \(\triangle ABC\) and \(\triangle ACD\) to get \(\frac{AB}{AC}=\frac{AC}{AD}\), and for \(\triangle ABC\) and \(\triangle CBD\) to get \(\frac{AB}{BC}=\frac{BC}{DB}\).