# 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 :

(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.