Question

If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, proves that $\frac{AB}{PQ}=\frac{AD}{PM}$

Answer 1

Given: AD and PM are medians of triangles ABC and PQR
ΔABC ~ ΔPQR

**To Prove: **$\frac{AB}{PQ}=\frac{AD}{PM}$

Proof: It is given that ∆ABC ∼ ∆PQR

We know that the corresponding sides of similar triangles are in proportion.
∴$\frac{AB}{PQ}=\frac{AC}{PR}=\frac{BC}{QR}$ … (1)
Also, $\angle$A = $\angle$P, $\angle$B = $\angle$Q, $\angle$C = $\angle$R … (2)

Since AD and PM are medians, they will divide their opposite sides.
∴BD=$\frac{BC}{2}$ and QM=$\frac{QR}{2}$ … (3)

From equations (1) and (3), we obtain
$\frac{AB}{PQ}=\frac{BD}{QM}$ … (4)

In ∆ABD and ∆PQM,
$\angle$B = $\angle$Q [Using equation (2)]
$\frac{AB}{PQ}=\frac{BD}{QM}$[Using equation (4)]
∴ ∆ABD ∼ ∆PQM (By SAS similarity criterion)

⇒ $\frac{AB}{PQ}=\frac{BD}{QM}=\frac{AD}{PM}$

$\therefore\frac{AB}{PQ}=\frac{AD}{PM}$

Hence Proved