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# In three-line segments OA, OB, and OC, points L, M, N respectively are so chosen that LM ∥ AB and MN ∥ BC but neither of L, M, and N nor A, B, C are collinear. Show that LN ∥ AC.

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Question drawn from very renowned book RD sharma of class 10th, Chapter no. 4
Chapter name:- Triangles
Exercise :- 4.2
This is very important question of triangle.

In this question we have three-line segments OA, OB, and OC,  and points L, M, N respectively

And they are chosen that in such a way that LM ∥ AB and MN ∥ BC but neither of L, M, and N nor A, B, C are collinear.

Now we have to Show that LN ∥ AC by using triangle geometrical properties.

learning CBSE maths in efficient way

RD sharma, DHANPAT RAI publication
Class 10th, triangle

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1. Given :

OA, OB and OC, points are L, M, and N respectively

Such that LM || AB and MN || BC

To prove: LN ∥ AC

Now,

In ΔOAB, Since, LM ∥ AB,

Then, OL/LA = OM/ MB (By using BPT)           – equation 1

In ΔOBC, Since, MN ∥ BC

Then, OM/MB = ON/NC        (By using BPT)

Therefore, ON/NC = OM/ MB                                        – equation 2

From equation 1 and 2, we get

OL/LA = ON/NC

Therefore, In ΔOCA By converse BPT, we get

LN || AC

Hence proved

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