Question

Triangle ABC and triangle DBC lie on the same side of BC. If BA=CD,BD=CA. Prove that AD||BC​

Answer 1

Given :

• Two triangles ABC and DBC lie on the

same side of the base BC. Points P,Q and R are

points on BC,AC and CD respectively such that

PR||BD and PQ||AB.

To prove :

• QR||AD

Proof :

In △ABC, we have

In △ABC, we have PQ∣∣AB

$\\ \\ \\ \\ \\ \\by \: \: using \: : \\ \\ \\ \\ \\ by \: \: proportionality \: \: theorem :\\ \\ \\ \\ \\ \\ \frac{CP}{PB} = \frac{CQ}{QA} \: \: - - - - - (i) \:$

In △BCD, we have

PR∣∣BD

$\\ \\ \\ \\ by \: \: using : \\ \\ \\ \\ \\ \\thale' s \: \: \: \: theorem \: \: : \: \\ \\ \\ \\ \frac{CP}{PB} = \frac{CR}{RD}$

From (i) and (ii), we have

BY BASIC PROPORTIONALITY THEOREM ,

$\\ \\ \\ \\ \\ \frac{CQ }{QA} = \frac{CR}{RD}$

Thus, in △ACD, Q and R are points on AC and CD

respectively such that

$\\ \\ \\ \\ \frac{CQ }{QA} = \frac{CR}{RD}$

So , QR∣∣AD [By the converse of Basic Proportionality Theorem]