Question

given right triangle abc with altitude \\(\overline{bd}\\) drawn to hypotenuse \\(\overline{ac}\\). if \\(ad = 27\\) and \\(dc = 3\\), what is the length of \\(\overline{bd}\\)?

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem)

In a right triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments into which it divides the hypotenuse. So, \( BD^2 = AD \times DC \).

Step2: Substitute the given values

We know \( AD = 27 \) and \( DC = 3 \). Substituting these into the formula, we get \( BD^2 = 27 \times 3 \).

Step3: Calculate the product and take the square root

First, calculate \( 27 \times 3 = 81 \). Then, take the square root of 81 to find \( BD \): \( BD = \sqrt{81} = 9 \).

Answer:

The length of \( \overline{BD} \) is \( 9 \).