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Question
If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that `("AB")/("PQ") = ("AD")/("PM")`.
Solution
Given, ΔABC ∼ ΔPQR
⇒ `("AB")/("PQ") = ("BC")/("QR") = ("AC")/("PR")`
(From the side-ratio property of similar triangles)
⇒ ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R ...(i)
BC = 2BD and QR = 2QM ...(∵ P and M are the midpoints of BC and QR)
⇒ `("AB")/("PQ") = (2 "BD")/(2 "QM") = ("AC")/("PR")`
⇒ `("AB")/("PQ") = ("BD")/("QM") = ("AC")/("PR")` ...(ii)
Now in ΔABD and ΔPQM
`("AB")/("PQ") = ("BD")/("QM")` ...(From (ii))
∠B = ∠Q ...(From (i))
⇒ ΔABD ∼ ΔPQM ...(By SAS similarity criterion)
∴ `("AB")/("PQ") = ("AD")/("PM")`
Hence, proved.
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