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# M and N are the points on the sides PQ and PR respectively, of a ΔPQR. For each of the following cases, state whether MN ∥ QR. (i) PM = 4 cm, QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm. (ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm.

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Question extracted from RD sharma of class 10th, Chapter no. 4
Chapter name:- Triangles
Exercise :- 4.2
This is very basic and important questions. And has been asked in several times in exams.

In this question we have question in two parts

It is given that M and N are the points on the sides PQ and PR respectively, of a ΔPQR.

Now we have to state whether MN ∥ QR.

Fow following two cases (i) PM = 4 cm, QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm.

And second case is (ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm.

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1. (i)

Given:

In the ∆PQR

M and N are points on PQ and PR respectively

PM = 4 cm, QM = 4.5 cm, PN = 4 cm, RN = 4.5 cm

To find: MN || QR or not

According to Thales Theorem,

PM/QM = PN/NR

4/4.5 = 4/4.5

Hence, MN || QR

(ii)

Given :

Length of side PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, and PN = 0.32 cm.

To find: that MN ∥ QR or not.

MQ = PQ – PM = 1.28 – 0.16 = 1.12 cm

RN = PR – PN = 2.56 – 0.32 =2.24 cm

According to Thales Theorem,

PM/QM = PN/ RN

PM/QN = 0.16/1.12 = 1/7                                              – equation 1

PN/RN = 0.32/ 2.24 = 1/7                                              – equation 2

From equation 1 and 2

PM/QM = PN/ RN

Therefore, MN || QR

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