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 Pythagorean theorem to find BC

In right - triangle \(ABC\) with \(\angle B = 90^{\circ}\), by the Pythagorean theorem \(AC^{2}=AB^{2}+BC^{2}\). Substitute \(AB = 15\) and \(AC = 38\) into the formula: \(BC=\sqrt{AC^{2}-AB^{2}}=\sqrt{38^{2}-15^{2}}=\sqrt{(38 + 15)(38 - 15)}=\sqrt{53\times23}\). Also, the area of right - triangle \(ABC\) can be expressed in two ways: \(S=\frac{1}{2}AB\times BC=\frac{1}{2}AC\times BD\). So \(AB\times BC=AC\times BD\), and \(\frac{BC}{BD}=\frac{AC}{AB}\).

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

Since \(AC = 38\) and \(AB = 15\), then \(\frac{BC}{BD}=\frac{38}{15}\).

Answer:

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