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CAT 2023 Slot 1 · Quant · Coordinate Geometry · medium

In triangle ABC, D is the midpoint of BC and E is the midpoint of AD. Line BE, when extended, meets AC at F. Find the ratio AF:FC.

Single correct option

In triangle ABC, D is the midpoint of BC and E is the midpoint of AD. Line BE, when extended, meets AC at F. Find the ratio AF:FC.
  1. A. 1:2
  2. B. 1:3
  3. C. 2:3
  4. D. 1:4
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Answer: (A) 1:2

Place A=(0,4), B=(0,0), C=(6,0). Then D (midpoint of BC) = (3,0), and E (midpoint of AD) = (1.5, 2). The line BE has direction (1.5,2); parametrize as (1.5t, 2t). The line AC goes from (0,4) to (6,0); parametrize as (6s, 4−4s). Equating: 1.5t = 6s and 2t = 4−4s. From the first, t = 4s; substituting: 8s = 4−4s ⇒ s = 1/3. Since s measures the fraction of AC from A, AF:AC = 1:3, so AF:FC = 1:2.
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Original question written in the CAT 2023 Slot 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake