Question

given right triangle abc with altitude \\(\overline{bd}\\) drawn to hypotenuse \\(\overline{ac}\\). if \\(bc = 2\\) and \\(dc = 1\\), what is the length of \\(\overline{ac}\\)?

Explanation:

Step1: Apply geometric mean theorem

In a right triangle, the length of a leg is the geometric mean of the hypotenuse and the adjacent segment. For leg $\overline{BC}$, this gives:
$$BC^2 = DC \times AC$$

Step2: Substitute known values

Replace $BC=2$ and $DC=1$ into the formula:
$$2^2 = 1 \times AC$$

Step3: Solve for $AC$

Calculate the square and isolate $AC$:
$$4 = AC$$

Answer:

$4$