Question

question 9 of 10
what is the value of h in the figure below? in this diagram, $\triangle bad \sim \triangle cbd$.
image of a right triangle with altitude h, base ac = 20, dc = 4
a. $\sqrt{80}$
b. 16
c. $\frac{16}{5}$
d. 80
e. 5
f. 8

Explanation:

Step1: Find length of AD

The total length of AC is 20, and CD is 4. So \( AD = AC - CD = 20 - 4 = 16 \).

Step2: Use similarity of triangles

Since \( \triangle BAD \sim \triangle CBD \), the ratios of corresponding sides are equal. That is \( \frac{AD}{BD}=\frac{BD}{CD} \). Let \( BD = h \), so \( \frac{16}{h}=\frac{h}{4} \).

Step3: Solve for h

Cross - multiply the proportion \( \frac{16}{h}=\frac{h}{4} \) to get \( h^{2}=16\times4 = 64 \)? Wait, no, wait. Wait, AD is 16? Wait, no, wait the length of AC is 20, and CD is 4, so AD is \( 20 - 4=16 \)? Wait, no, maybe I made a mistake. Wait, the segment from A to O is AD? Wait, no, the diagram shows that AC is 20, CD is 4, so AD is \( 20 - 4 = 16 \)? Wait, but when we have similar triangles \( \triangle BAD \sim \triangle CBD \), the corresponding sides: in \( \triangle BAD \), the sides are AD, BD, AB; in \( \triangle CBD \), the sides are CD, BD, BC. Wait, actually, the correct proportion is \( \frac{AD}{BD}=\frac{BD}{CD} \), so \( BD^{2}=AD\times CD \). Wait, AD is \( 20 - 4 = 16 \)? Wait, no, maybe the length of AD is \( 20 - 4=16 \)? Wait, no, let's re - examine. The total length of AC is 20, CD is 4, so AD is \( 20 - 4 = 16 \). Then \( h^{2}=AD\times CD=16\times4 = 64 \)? No, that would give \( h = 8 \). Wait, that's different from my initial thought. Wait, maybe I messed up the segments. Wait, maybe the altitude to the hypotenuse of a right triangle is the geometric mean of the segments into which it divides the hypotenuse. In a right triangle, if we have a right triangle ABC with right angle at B, and BD is the altitude to AC, then \( \triangle BAD \sim \triangle CBD \sim \triangle ABC \), and \( BD^{2}=AD\times CD \). So if AC is 20, CD is 4, then AD is \( 20 - 4 = 16 \). Then \( h^{2}=16\times4=64 \), so \( h = 8 \).

Answer:

F. 8