Question

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. (Recall that you have done it in Class IX) .

Answer 1

Given that

A triangle with mid points

To prove

The line joining the mid-points of any two sides of a triangle is parallel to the third side.

Proof

We have to prove that: $\text{DE || BC}$.

Since, $\text{D}$ is the midpoint of $\text{AB}$

$\begin{array}{rcl}& & \text{∴ AD = DB}\\ & & \text{⇒}\frac{\text{AD}}{\mathrm{BD}}\text{= 1……………………. (i)}\end{array}$

Also given, $\text{E}$ is the mid-point of $\text{AC}$.

$$\begin{gathered} \therefore AE=EC \\ \Rightarrow \frac{AE}{EC}=1 \end{gathered}$$

From equation (i) and (ii),

we get, $\frac{\mathrm{AD}}{\mathrm{BD}}=\frac{\mathrm{AE}}{\mathrm{EC}}$

On applying the converse of Basic Proportionality Theorem,

$\mathrm{DE}\left|\right|\mathrm{BC}$

Hence, proved.