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