Ex.6.2 Q8 Triangles Solution - NCERT Maths Class 10

Using Theorem \(6.2\), 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).

Text Solution

Reasoning:

We know that theorem \(6.2\) tells us if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. (Converse of \(BPT\))

Steps:

In \(\Delta ABC\)

\(D\) is the midpoint of \(AB\)

\[\begin{align}AD &= BD\\  \frac{AD}{BD} &= 1\;\dots\rm{(i)}\end{align}\]

\(E\) is the midpoint of \(AC\)

\[\begin{align} AE &= CE\\ \frac{AE}{BE} &= 1\;\dots\rm{(ii)}\end{align}\]

From (i) and (ii)

\[\begin{align} \frac{AD}{BD}&=\frac{AE}{BE} = 1 \\ \frac{AD}{BD}&=\frac{AE}{BE} \\\end{align}\]

According to theorem \(6.2,\) (Converse of \(BPT\))

\[DE||BC\]