Question

given is the right triangle abc with altitude bd. apply the pythagorean theorem and similar triangles as needed to find the length of \overline{ad}. (1 point) \frac{289}{15} \frac{136}{15} 17 \frac{64}{15}

Explanation:

Step1: Use the geometric mean theorem

In a right - triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Also, we know that in right - triangle \(ABD\) and \(ABC\), \(\triangle ABD\sim\triangle ABC\) and \(\triangle BCD\sim\triangle ABC\) and \(\triangle ABD\sim\triangle BCD\). The geometric mean theorem states that \(BD^{2}=AD\times DC\). Let \(AD = x\), then \(DC = 15\), and \(BD = 8\). So, \(8^{2}=x\times15\).

Step2: Solve for \(x\)

We have the equation \(64 = 15x\). Then \(x=\frac{64}{15}\)

Answer:

\(\frac{64}{15}\)