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Line parallel to a side cuts the other sides in proportion: find a segment

This type of question, asked 3 times: CDS 2 2017, CDS 1 2021, CDS 2 2026. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2017

    77
    In a triangle ABC, the medians AD and BE intersect at G. A line DF is drawn parallel to BE such that F is on AC. If AC = 9 cm, then what is CF equal to ?

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  2. Question 2

    CDS 1 2021

    82
    ABCABC is a triangle right angled at BB. Let MM and NN be two points on ABAB such that AM=MN=NBAM = MN = NB. Let PP and QQ be two points on ACAC such that PMPM is parallel to QNQN and QNQN is parallel to CBCB. If BC=12BC = 12 cm, then what is (PM+QN)(PM + QN) equal to?

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  3. Question 3

    CDS 2 2026

    70
    In a triangle ABC, AB = 8 cm and D and E are points on AB and AC respectively such that DE is parallel to BC. If BC = 5DE, then what is AD ×\times BD equal to ?

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