Question

Question 8
Using converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.

Answer 1

Given, $\Delta ABC$ in which D and E are the mid points of AB and AC respectively such that AD=BD and AE=EC.
To Prove that DE || BC
Proof
D is the mid-point of AB (Given)
$\therefore AD=DB$
$\Rightarrow \frac{AD}{BD}=1...\left(i\right)$
Also, E is the mid-point of AC (Given)
$\therefore AE=EC$
$\Rightarrow \frac{AE}{EC}=\mathrm{1.....}\left(ii\right)$
From equation (i) and (ii), we get
$\frac{AD}{BD}=\frac{AE}{EC}$, hence, DE || BC [By converse of Basic Proportionality Theorem]