Question

in the figure below, there are three right triangles. complete the following. (a) write a similarity statement relating the three right triangles. $\triangle abc \sim \triangle \square \sim \triangle \square$ (b) complete each proportion. $\dfrac{bd}{bc} = \dfrac{cb}{\square}$ $\dfrac{ad}{cd} = \dfrac{\square}{bd}$

Explanation:

Step1: Identify similar triangles

In right $\triangle ABC$ with altitude $CD$ to hypotenuse $AB$, the three right triangles are similar by AA similarity (shared angles and right angles). So $\triangle ABC \sim \triangle CBD \sim \triangle ACD$.

Step2: Solve first proportion

From $\triangle ABC \sim \triangle CBD$, corresponding sides give $\frac{BD}{BC} = \frac{CB}{AB}$.

Step3: Solve second proportion

From $\triangle ACD \sim \triangle CBD$, corresponding sides give $\frac{AD}{CD} = \frac{CD}{BD}$.

Answer:

(a) $\triangle ABC \sim \triangle CBD \sim \triangle ACD$
(b) $\frac{BD}{BC} = \frac{CB}{AB}$; $\frac{AD}{CD} = \frac{CD}{BD}$