OG详解-OG6 数学1 Q27

The correct answer is 54. It’s given that in triangle ABC, 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. It follows that parallel segments DE and BC are intersected by sides AB and AC. If two parallel segments are intersected by a third segment, corresponding angles are congruent. Thus, corresponding angles C and AED are congruent and corresponding angles B and ADE are congruent. Since triangle ADE has two angles that are each congruent to an angle in triangle ABC, triangle ADE is similar to triangle ABC by the angle-angle similarity postulate, where side DE corresponds to side BC, and side AE corresponds to side AC. Since the lengths of corresponding sides in similar triangles are proportional, it follows that DEBC​=AEAC​​. Since point E lies on side AC, AE+CE = AC. It’s given that CE = 2AE. Substituting 2AE for CE in the equation AE+CE = AC yields AE+2AE = AC, or 3AE = AC. It’s given that BC = 162. Substituting 162 for BC and 3AE for AC in the equation DEBC​=AEAC​​ yields DE162​=AE3​AE​ or  DE162​=13​​. Multiplying both sides of this equation by 162 yields DE = 54. Thus, the length of line segment DE is 54.