Sum

AD and PM are medians of triangles ABC and PQR, respectively where Δ ABC ~ Δ PQR, prove that `("AB")/("PQ") = ("AD")/("PM")`

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#### Solution

It is given that ΔABC ∼ ΔPQR

We know that the corresponding sides of similar triangles are in proportion.

`:.("AB")/("PQ")=("AC")/"AD" and =("BC")/("QR")....(1)`

Also, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R … (2)

Since AD and PM are medians, they will divide their opposite sides.

`:."BD" = ("BC")/2 ` and `"QM" = "QR"/2`...(3)

From equations (1) and (3), we obtain

`("AB")/("PQ") = ("BD")/("QM") ....(4)`

In ΔABD and ΔPQM,

∠B = ∠Q [Using equation (2)]

`("AB")/("PQ") =("BD")/("QM") `

∴ ΔABD ∼ ΔPQM (By SAS similarity criterion)

`=> ("AB")/("PQ") = ("BD")/("QM") = ("AD")/("PM")`

Concept: Criteria for Similarity of Triangles

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