SAT Math: In Right Triangle ABC, Altitude CD is Drawn to Hypotenuse AB (AD=4, DB=16)

Exact Question Text

In right triangle $\triangle ABC$, angle $C$ is a right angle. Altitude $\overline{CD}$ is drawn to the hypotenuse $\overline{AB}$.

If $AD = 4$ and $DB = 16$, what is the length of $\overline{CD}$?

Step-by-Step Solution

1. Apply Geometric Mean (Altitude) Theorem:

Drawing altitude $CD$ creates two similar sub-triangles $\triangle ADC \sim \triangle CDB$:

$$ \frac{AD}{CD} = \frac{CD}{DB} \implies (CD)^2 = AD \cdot DB $$

1. Substitute Values:

$$ (CD)^2 = 4 \cdot 16 = 64 \implies CD = \sqrt{64} = \mathbf{8} $$