Numerical answer
In triangle ABC, points D and E lie on AB and AC respectively such that DE is parallel to BC and AD : DB = 2 : 3. If the area of the quadrilateral DBCE is 84 sq. units, what is the area of triangle ABC (in sq. units)?
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Answer: 100
Since DE ∥ BC, triangle ADE ~ triangle ABC with ratio AD : AB = 2 : 5.
Area ratio = (2/5)² = 4/25. So ADE is 4/25 of ABC and the trapezium DBCE is 21/25 of ABC.
21/25 × [ABC] = 84 ⇒ [ABC] = 100 sq. units.
Trap: using AD : DB = 2 : 3 directly as the similarity ratio — it must be AD : AB.
Area ratio = (2/5)² = 4/25. So ADE is 4/25 of ABC and the trapezium DBCE is 21/25 of ABC.
21/25 × [ABC] = 84 ⇒ [ABC] = 100 sq. units.
Trap: using AD : DB = 2 : 3 directly as the similarity ratio — it must be AD : AB.
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