Question

given right triangle abc with altitude \\(\overline{bd}\\) drawn to hypotenuse \\(\overline{ac}\\). if \\(bd = 15\\) and \\(dc = 25\\), what is the length of \\(\overline{ad}\\)?

Explanation:

Step1: Apply geometric mean theorem

For a right triangle, the altitude to the hypotenuse satisfies $BD^2 = AD \times DC$.

Step2: Isolate $AD$

Rearrange the formula to solve for $AD$: $AD = \frac{BD^2}{DC}$

Step3: Substitute given values

Substitute $BD=15$ and $DC=25$:
$AD = \frac{15^2}{25} = \frac{225}{25}$

Step4: Simplify the expression

Calculate the division: $AD = 9$

Answer:

$9$