In right triangle ABC, angle C is the right angle, and BC = 162. Point D on side AB is connected by a line segment with point E on side AC such that line segment DE is parallel to side BC and CE = 2AE. What is the length of line segment DE?

Explanation

The correct answer is 54. To solve this problem, recognize that triangle ADE is similar to triangle ABC because DE is parallel to BC, making corresponding angles congruent. Given CE = 2AE, we can express AC in terms of AE: AC = AE + CE = 3AE. Since DE/BC = AE/AC, substituting the known values gives DE/162 = AE/3AE = 1/3. Solving this proportion by multiplying both sides by 162 gives DE = 54. Other options result from misapplying the proportional relationships or incorrect calculations of the side lengths.