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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). Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Solution:

Given: In △ABC, D and E are midpoints of AB and AC respectively, i.e., AD=DB and AE=EC
To Prove: DE∥BC
Proof:Since, AD=DB ∴ AD/DB=1 (1)
Also, AE=EC ∴ AE/EC=1 (2)
From (1) and (2), AD/DB = AE/EC = 1
i.e., AD/DB = AE/EC
∴ By converse of Basic Proportionality theorem, DE∥BC