Question

part 1 of 2
use the figure shown to the right to prove the following.
given: right △abc with altitude to the hypotenuse cd
prove: $\frac{ab}{ac}=\frac{ac}{ad}$, $\frac{ab}{bc}=\frac{bc}{db}$

statements reasons

1. right △abc with altitude to the hypotenuse cd. 1. given
2. 2. altitude of a right triangle theorem

Explanation:

Step1: Identify similar triangles

In right - triangle $\triangle ABC$ with altitude $CD$ to the hypotenuse $AB$, $\triangle ACD\sim\triangle ABC$ and $\triangle BCD\sim\triangle BAC$.
For $\triangle ACD\sim\triangle ABC$, the ratios of corresponding sides are equal. That is, $\frac{AB}{AC}=\frac{AC}{AD}$ because in similar triangles, the ratios of corresponding sides are the same.

Step2: Consider other similar triangles

For $\triangle BCD\sim\triangle BAC$, we have $\frac{AB}{BC}=\frac{BC}{DB}$ since the corresponding - side ratios of similar triangles are equal.

Answer:

The proof is completed by showing that $\frac{AB}{AC}=\frac{AC}{AD}$ and $\frac{AB}{BC}=\frac{BC}{DB}$ using the property of similar right - triangles formed by the altitude to the hypotenuse of a right - triangle.