Question

in triangle abc, the measure of angle b is 90° 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 \(AC=AB + 23\), so \(AC=15 + 23=38\).

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

In right - triangle \(ABC\) with \(\angle B = 90^{\circ}\) and altitude \(BD\), we know that \(AB\times BC=AC\times BD\) (by the area formula \(S=\frac{1}{2}AB\times BC=\frac{1}{2}AC\times BD\)). Then \(\frac{BC}{BD}=\frac{AC}{AB}\).

Step3: Calculate the ratio \(\frac{BC}{BD}\)

Substitute \(AB = 15\) and \(AC = 38\) into \(\frac{BC}{BD}=\frac{AC}{AB}\), we get \(\frac{BC}{BD}=\frac{38}{15}\).

Answer:

D. \(\frac{38}{15}\)