Question

Using theorem 6.1 prove that a line drawn through the midpoint of one side of a triangle is parallel to the third side

Answer 1

Solution: In Δ PQO and Δ PED;

similar triangles exercise solution

(Because these are similar triangles, as per Basic Proportionality theorem.)

Similarly, in Δ PRO and Δ PFD;

similar triangles exercise solution

From above two equations, it is clear;

similar triangles exercise solution

Hence;

similar triangles exercise solution

Hence, EF || QR proved.

Answer 2

Given,In triangle ABC, D is the midpoint of AB such that AD=DB.

A line parallel to BC intersects AC at E as shown in above figure such that DE||BC.

To prove, E is the midpoint of AC.

Since, D is the midpoint of AB

So,AD=DB

⇒ AD/DB=1.....................(i)

In triangle ABC,DE||BC,

By using basic proportionality theorem,

Therefore, AD/DB=AE/EC

From equation 1,we can write,

⇒ 1=AE/EC

So,AE=EC

Hence, proved,E is the midpoint of AC.