Question

1. in △abc, ∠acb = 90°, and cd is the altitude to ab. if ac = 5, bc = 12, and ab = 13, what is the length of cd? a. 5.7 b. 4.5 c. 6.9 d. 4.6

Explanation:

Step1: Use area - formula of right - triangle

The area of right - triangle $\triangle ABC$ can be calculated in two ways. First, $S=\frac{1}{2}AC\times BC$ since $\angle ACB = 90^{\circ}$. Second, $S=\frac{1}{2}AB\times CD$ where $CD$ is the altitude to the hypotenuse $AB$.

Step2: Equate the two area - expressions

We know that $\frac{1}{2}AC\times BC=\frac{1}{2}AB\times CD$. Given $AC = 5$, $BC = 12$, and $AB = 13$. Substitute the values into the equation: $5\times12=13\times CD$.

Step3: Solve for $CD$

$CD=\frac{5\times12}{13}=\frac{60}{13}\approx4.6$

Answer:

d. 4.6