Question

1. in △abc, ∠acb = 90°, and cd is the altitude to ab. which of the following is true? a. △acd and △bcd are congruent to each other b. △acd ~ △abc ~ △bcd c. the altitude divides ab into two congruent segments d. cd is proportional to the sum of ac and bc

Explanation:

Step1: Recall right - triangle similarity

In right - triangle $\triangle ABC$ with right - angle at $\angle ACB$ and altitude $CD$ to hypotenuse $AB$, we use the geometric mean theorem. The three right - triangles $\triangle ACD$, $\triangle BCD$, and $\triangle ABC$ are similar.
We know that $\angle ADC=\angle BDC = \angle ACB=90^{\circ}$, and $\angle A+\angle B = 90^{\circ}$, $\angle A+\angle ACD = 90^{\circ}$, so $\angle B=\angle ACD$ and $\angle A=\angle BCD$. By the AA (angle - angle) similarity criterion, $\triangle ACD\sim\triangle ABC\sim\triangle BCD$.

Step2: Analyze each option

• Option a: $\triangle ACD$ and $\triangle BCD$ are not congruent. They are similar because $\angle ADC=\angle BDC = 90^{\circ}$, $\angle A=\angle BCD$ and $\angle B=\angle ACD$, but their side - lengths are not equal in general.
• Option b: Since $\angle ADC=\angle BDC=\angle ACB = 90^{\circ}$, $\angle A=\angle BCD$ and $\angle B=\angle ACD$, by AA similarity, $\triangle ACD\sim\triangle ABC\sim\triangle BCD$. This option is correct.
• Option c: The altitude $CD$ does not divide $AB$ into two congruent segments. In general, $AD

eq BD$ unless $\triangle ABC$ is an isosceles right - triangle.

• Option d: There is no such proportionality $CD$ is proportional to the sum of $AC$ and $BC$.

Answer:

b. $\triangle ACD\sim\triangle ABC\sim\triangle BCD$