Ex 6.3, 12 - Sides AB and BC and median AD of a ABC - Ex 6.3

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Ex 6.3, 12 Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see figure). Show that ΔABC ∼ ΔPQR. Given: ΔABC where AD is the median Δ PQR where PM is the median & 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅=𝐴𝐷/𝑃𝑀 To Prove: ΔABC ∼ ΔPQR. Proof:- Since AD is the median, BD = CD = 1/2 BC Similarly, PM is the median, QM = RM = 1/2QR Given that 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅=𝐴𝐷/𝑃𝑀 𝐴𝐵/𝑃𝑄=2𝐵𝐷/2𝑄𝑀=𝐴𝐷/𝑃𝑀 𝐴𝐵/𝑃𝑄=𝐵𝐷/𝑄𝑀=𝐴𝐷/𝑃𝑀 Since all 3 sides are proportional ΔABD ∼ ΔPQM Hence, ∠𝐵=∠𝑄 In Δ ABC & ΔPQR ∠𝐵=∠𝑄 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅 Hence by SAS similarly ΔABC ∼ ΔPQR

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