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CAT 2016 Slot 2 · Quant · Coordinate Geometry · hard

E is the midpoint of AB and F is the midpoint of BC. The segments DE and AF intersect at G. What is the area of triangle ADG (in cm²)?

Single correct option

ABCD is a square of side 10 cm. E is the midpoint of AB and F is the midpoint of BC. The segments DE and AF intersect at G. What is the area of triangle ADG (in cm²)?
  1. A. 16
  2. B. 20
  3. C. 25
  4. D. 30
Show answer and solution

Answer: (B) 20

Place A(0, 0), B(10, 0), C(10, 10), D(0, 10). Then E = (5, 0) and F = (10, 5).

Line DE: through (0, 10) and (5, 0), so y = 10 − 2x.
Line AF: through (0, 0) and (10, 5), so y = x/2.

Intersection: x/2 = 10 − 2x gives x = 4, y = 2. So G = (4, 2).

Triangle ADG has base AD = 10 on the y-axis, and its height is the x-coordinate of G, which is 4.
Area = ½ × 10 × 4 = 20.

Note: DE ⟂ AF (slopes −2 and 1/2). This is the well-known perpendicular-segments configuration in a square.
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Original question written in the CAT 2016 Slot 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake