Question

in the diagram below, \\(\overline{bc}\\) is an altitude of \\(\triangle abd\\). to the nearest whole unit, what is the length of \\(cd\\)? \\(\bigcirc\\) a. 24 \\(\bigcirc\\) b. 26 \\(\bigcirc\\) c. 21 \\(\bigcirc\\) d. 29

Explanation:

Step1: Identify similar triangles

Since $\overline{BC}$ is an altitude of right $\triangle ABD$ (right-angled at $B$), $\triangle BCD \sim \triangle ACB$.

Step2: Set up proportion for similarity

For similar triangles, $\frac{CD}{BC} = \frac{BC}{AC}$.
Substitute $BC=37$, $AC=53$:
$\frac{CD}{37} = \frac{37}{53}$

Step3: Solve for $CD$

Rearrange to isolate $CD$:
$CD = \frac{37^2}{53} = \frac{1369}{53}$

Step4: Calculate and round

$\frac{1369}{53} \approx 25.83$, which rounds to 26.

Answer:

B. 26