Question

in right triangle abc, cd is the altitude to hypotenuse ab. if ad = 2 and db = 18, find cd.

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem) for right triangles. The theorem states that in a right triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments into which it divides the hypotenuse. Mathematically, this means \( CD^2 = AD \times DB \).

Step2: Substitute the given values of \( AD = 2 \) and \( DB = 18 \) into the formula. So we have \( CD^2 = 2 \times 18 \).

Step3: Calculate the product on the right side. \( 2 \times 18 = 36 \), so \( CD^2 = 36 \).

Step4: Take the square root of both sides to find \( CD \). Since \( CD \) represents a length (and thus is non - negative), \( CD=\sqrt{36}=6 \).

Answer:

\( CD = 6 \)