In △ABC, a line segment DE is parallel to BC, with D on AB and E on AC. If the ratio of the area of △ADE to the area of the trapezoid DECB is 4:21, what is the ratio of AD to DB?

Correct — A. DE ∥ BC makes △ADE similar to △ABC, and similar areas go as the square of the side ratio.

Add the parts: △ADE : △ABC = 4 : (4 + 21) = 4 : 25
Take square roots: AD : AB = 2 : 5
Subtract: DB = AB − AD = 5 − 2 = 3 parts
So AD : DB = 2 : 3 → option (a)

Why the others are wrong

• (b)
  2:5 — 2 : 5 is AD : AB, the part against the whole side. The question asks AD : DB, and DB is the 3 parts of AB left after AD's 2.
• (c)
  3:5 — 3 : 5 is DB : AB. It measures the lower part against the whole side, not AD against DB.
• (d)
  4:5 — 4 : 5 would make AD : AB = 4 : 9, so △ADE : △ABC = 16 : 81 and △ADE : DECB = 16 : 65, not 4 : 21.

A line parallel to one side of a triangle, cutting the other two, cuts off a smaller similar triangle: the corresponding angles are equal.

Similar triangles have areas in the ratio of the squares of corresponding sides, so a side ratio of k gives an area ratio of k².

The trapezoid is the big triangle minus the small one. Turn the part-to-part ratio 4 : 21 into part-to-whole, 4 : 25, before taking the square root.

Key facts

• DE ∥ BC gives △ADE ~ △ABC (equal corresponding angles).
• Basic proportionality theorem: DE ∥ BC gives AD⁄DB = AE⁄EC.
• Area ratio of similar triangles = (side ratio)².
• Trapezoid DECB = △ABC − △ADE.

Common traps

• Taking the square root of 4 : 21 directly. The similarity is with the whole triangle ABC, so the ratio must become 4 : 25 first.
• Stopping at AD : AB = 2 : 5 when the question asks for AD : DB.

No directly related past PYQ was found.