Ex.6.2 Q8 Triangles Solution - NCERT Maths Class 10
Question
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 \(AB\)
\[AD = BD\]
\[\begin{align}\frac{AD}{BD} = 1\,\,............\rm{(i)}\end{align}\]
\(E\) is the midpoint of \(AC\)
\[AE = CE\]
\[\begin{align}\frac{AE}{BE} = 1............\rm{(ii)}\end{align}\]
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 \(BPT\))
\[DE||BC\]