Question

given right triangle abc with altitude \\(\overline{bd}\\) drawn to hypotenuse \\(\overline{ac}\\). if \\(ad = 8\\) and \\(ac = 32\\), what is the length of \\(\overline{ab}\\)?

Explanation:

Step1: Apply geometric mean theorem

In a right triangle, the length of a leg is the geometric mean of the length of the hypotenuse and the length of the adjacent segment of the hypotenuse. So:
$$AB^2 = AD \times AC$$

Step2: Substitute given values

Replace $AD=8$ and $AC=32$ into the formula:
$$AB^2 = 8 \times 32$$

Step3: Calculate and solve for AB

First compute the product, then take the square root:
$$AB^2 = 256$$
$$AB = \sqrt{256}$$

Answer:

16