Question

given right triangle abc with altitude \\(\overline{bd}\\) drawn to hypotenuse \\(\overline{ac}\\). if \\(ad = 10\\) and \\(bd = 30\\), what is the length of \\(\overline{dc}\\)?

Explanation:

Step1: Apply geometric mean theorem

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

Step2: Isolate $DC$

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

Step3: Substitute given values

Plug in $AD=10$ and $BD=30$:
$DC = \frac{30^2}{10} = \frac{900}{10}$

Step4: Calculate the result

Simplify the fraction to find $DC$.

Answer:

90