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 \(AC\) is 23 greater than \(AB\), so \(AC=AB + 23=15 + 23=38\).

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 \(\triangle ABC\sim\triangle ABD\sim\triangle BCD\). Also, by the geometric - mean theorem, \(AB^{2}=AD\times AC\) and \(BC^{2}=CD\times AC\), and \(BD^{2}=AD\times CD\). Another property is that \(\frac{BC}{BD}=\frac{AC}{AB}\).

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

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

Answer:

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