Question

given right triangle abc with altitude \\( \overline{bd} \\) drawn to hypotenuse \\( \overline{ac} \\). if \\( ad = 4 \\) and \\( bd = 2 \\), what is the length of \\( dc \\)?

Explanation:

Step1: Recall geometric theorem

In a right triangle, the altitude to the hypotenuse satisfies the geometric mean theorem: \( BD^2 = AD \times DC \).

Step2: Substitute known values

We know \( AD = 4 \), \( BD = 2 \). Substitute into the formula: \( 2^2 = 4 \times DC \).

Step3: Solve for DC

Simplify the equation: \( 4 = 4 \times DC \). Divide both sides by 4: \( DC = \frac{4}{4} = 1 \).

Answer:

1