Question

given right triangle abc with altitude \\( \overline{bd} \\) drawn to hypotenuse \\( \overline{ac} \\). if \\( bd = 24 \\) and \\( dc = 8 \\), what is the length of \\( \overline{ac} \\)?

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. That is \(BD^{2}=AD\times DC\).
We know \(BD = 24\) and \(DC=8\). Substitute the values into the formula: \(24^{2}=AD\times8\).

Step2: Solve for \(AD\)

From \(24^{2}=AD\times8\), we have \(AD=\frac{24^{2}}{8}\).
Since \(24^{2}=576\), then \(AD=\frac{576}{8}=72\).

Step3: Calculate the length of \(AC\)

Since \(AC=AD + DC\), and \(AD = 72\), \(DC = 8\).
So \(AC=72 + 8\).

Answer:

\(80\)