Question

In the figure shown, ABC is a right triangle with side lengths a, b, and c, and CD is an altitude to side AB. The side lengths of triangle ACD are b, h, and r, and the side lengths of triangle CBD are a, s, and h.

Which proportions are true?


A) c/a = a/s and c/b = b/r


B) c/a = s/a and c/b = r/b


C) c/b = a/s and c/a = b/r


D) c/b = s/a and c/a = r/b

Answer 1

A

Explanation:

In the figure shown, ABC is a right triangle with side lengths a, b, and c, and CD is an altitude to side AB. This altitude divides the triangle into two right triangles ADC and BDC. In these triangles,

• 
  `\angle CBD\cong \angle ACD\cong \angle CBA`
• 
  `\angle DCB\cong \angle DAC\cong \angle BAC`

So,

`\triangle ABC\sim \triangle CBD\sim \triangle ACD`

1. From the similarity
`\triangle ABC\sim \triangle CBD,` you have

`(AB)/(BC)=(BC)/(BD)\\ \\(c)/(a)=(a)/(s)`

2. From the similarity
`\triangle ABC\sim \triangle ACD,` you have

`(AB)/(AC)=(AC)/(AD)\\ \\(c)/(b)=(b)/(r)`

Hence, option A is true