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Question

Using Theorem $$6.1,$$ prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.


Solution

Given:  
In $$\triangle\, ABC, D$$ is midpoint of $$AB$$ and $$DE$$ is parallel to $$BC.$$
$$\therefore$$ $$AD=DB$$
To prove: 
$$AE = EC$$
Proof: 
Since, $$DE \parallel BC$$
$$\therefore$$ By Basic Proportionality Theorem,
$$\dfrac{AD}{DB}=\dfrac{AE}{EC}$$
Since, $$AD = DB$$
$$\therefore$$ $$\dfrac{AE}{EC}=1$$
$$\therefore$$ $$AE=EC$$

493896_465423_ans_3c86422f09da4d0d9d4b82d59554959c.png

Mathematics
RS Agarwal
Standard X

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