E and F are points on the side PQ and PR respectively of a triangle PQR. For each of the following cases state whether EF || QR

Question :

E and F are points on the side PQ and PR respectively of a triangle PQR. For each of the following cases state whether EF || QR :

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(i)  PE = 3.9 cm,  EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii)  PE = 4cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii)  PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

Solution :

By converse of basic  proportionality theorem or Thales theorem that  if a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

(i)  Here, \(PE\over EQ\) = \(3.9\over 3\) = \(1.3\over 1\)

\(PF\over FR\) = \(3.6\over 2.4\) = \(3\over 2\) = 1.5

Thus, \(PE\over EQ\) \(\ne\) \(PF\over FR\)

No, EF is not parallel to QR.

(ii)  Here, \(PE\over EQ\) = \(4\over 4.5\)

\(PF\over FR\) = \(8\over 9\) = \(4\over 4.5\)

Thus, \(PE\over EQ\) = \(PF\over FR\)

Yes, EF is parallel to QR.

(iii)  Here, \(PQ\over PE\) = \(1.28\over 0.18\)

\(PR\over PF\) = \(2.56\over 0.36\) = \(1.28\over 0.18\)

Thus, \(PQ\over PE\) \(\ne\) \(PR\over PF\)

Yes, EF is parallel to QR.

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