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).
The line joining the mid-points of any two sides of a triangle is parallel to the third side.
Statement to prove: In any triangle, the line segment joining the mid-points of two sides is parallel to the third side.
Consider \(\triangle ABC\).
Let \(D\) be the mid-point of side \(AB\) and \(E\) be the mid-point of side \(AC\).
So,
\[AD = DB \quad \text{and} \quad AE = EC.\]
Since \(D\) is the mid-point of \(AB\):
\[AD = DB \Rightarrow \frac{AD}{DB} = 1.\]
Since \(E\) is the mid-point of \(AC\):
\[AE = EC \Rightarrow \frac{AE}{EC} = 1.\]
Thus,
\[\frac{AD}{DB} = \frac{AE}{EC}.\]
Theorem 6.2: If a line intersects two sides of a triangle such that it divides those sides in the same ratio, then the line is parallel to the third side.
In \(\triangle ABC\), line segment \(DE\) cuts sides \(AB\) and \(AC\) at \(D\) and \(E\) respectively, and we have just shown
\[\frac{AD}{DB} = \frac{AE}{EC}.\]
Therefore, by Theorem 6.2,
\[DE \parallel BC.\]
The line joining the mid-points \(D\) and \(E\) of sides \(AB\) and \(AC\) is parallel to the third side \(BC\). Hence, the line joining the mid-points of any two sides of a triangle is parallel to the third side.