Question

in triangle (abc), the measure of angle (b) is (90^{circ}) and (overline{bd}) is an altitude of the triangle. the length of (overline{ab}) is 15 and the length of (overline{ac}) is 23 greater than the length of (overline{ab}). what is the value of (\frac{bc}{bd})?
a (\frac{15}{38})
b (\frac{15}{23})
c (\frac{23}{15})
d (\frac{38}{15})

Explanation:

Step1: Find the length of $AC$

Given $AB = 15$ and $AC$ is 23 greater than $AB$, so $AC=15 + 23=38$.

Step2: Use the geometric - mean theorem in right - triangle $ABC$

In right - triangle $ABC$ with right - angle at $B$ and altitude $BD$, we know that $AB\times BC=AC\times BD$. Then $\frac{BC}{BD}=\frac{AC}{AB}$.

Step3: Calculate the ratio $\frac{BC}{BD}$

Substitute $AC = 38$ and $AB = 15$ into the ratio $\frac{BC}{BD}=\frac{AC}{AB}$, we get $\frac{BC}{BD}=\frac{38}{15}$.

Answer:

D. $\frac{38}{15}$