Question

AD and PM are medians of triangles ABC and PQR respectively where triangle ABC ~ traingle PQR. prove that:-
$\frac{ab}{pq} = \frac{ad}{pm}$
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Answer 1

Answer:

It is given that ΔABC ~ ΔPQR

We know that the corresponding sides of similar triangles are in proportion.∴ AB/PQ = AC/PR = BC/QR ...(i)

Also, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R …(ii)

Since AD and PM are medians, they will divide their opposite sides.∴ BD = BC/2 and QM = QR/2 ...(iii)

From equations (i) and (iii), we get

AB/PQ = BD/QM ...(iv)

In ΔABD and ΔPQM,

∠B = ∠Q [Using equation (ii)]

AB/PQ = BD/QM [Using equation (iv)]

∴ ΔABD ~ ΔPQM (By SAS similarity criterion)⇒ AB/PQ = BD/QM = AD/PM

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