Question

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that : `("AB")/("PQ") = ("AD")/("PM")`.

Answer 1

Given, ΔABC ∼ ΔPQR; AD and PM are the medians of ΔABC and ΔPQR respectively.

To prove: `("AB")/("PQ") = ("AD")/("PM")`

Proof: Since, ΔABC ∼ ΔPQR

∴ ∠B = ∠Q and

`("AB")/("PQ") = ("BC")/("QR") = (2"BD")/(2"QM") = ("BD")/("QM")`   ...(∵ D and M are mid-points of BC and QR)

Now in ΔABD and ΔPQM

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

∠B = ∠Q   ...(Given)

∴ ΔABD ∼ ΔPQM    ...(SAS axiom of similarity)

∴ `("AB")/("PQ") = ("AD")/("PM")`   ...(Corresponding sides of Δ's are proportional)