E and F are points on the sides PQ and PR respectively of a triangle PQR. For each of the following cases, state whether EF || QR: (i) PE=3.9cm, EQ=3cm, PF=3.6cm and FR=2.4cm (ii) PE=4cm, QE=4.5cm, PF=8cm and RF=9cm (iii) PQ=1.28cm, PR=2.56cm, PE=0.18cm and PF=0.36cm

E and F are two points on side PQ and PR in triangle PQR
(i) PE = 3.9cm, EQ = 3cm and PF = 3.6cm, FR = 2.4cm
Using Basic proportionality theorem,
∴ PE/EQ = 3.9/3 = 39/30 = 13/10 = 1.3
PF/FR = 3.6/2.4 = 36/24 = 3/2 = 1.5
PE/EQ ≠ PF/FR
So, EF is not parallel to QR
(ii) PE = 4cm, QE = 4.5cm, PF = 8cm, RF = 9cm
Using Basic proportionality theorem,
∴ PE/QE = 4/4.5 = 40/45 = 8/9
PF/RF = 8/9
PE/QE = PF/RF
So, EF is parallel to QR
(iii) PQ = 1.28cm, PR = 2.56cm, PE = 0.18cm, PF = 0.36cm
Using Basic proportionality theorem,
EQ = PQ – PE = 1.28 – 0.18 = 1.10cm
FR = PR – PF = 2.56 – 0.36 = 2.20cm
PE/EQ = 0.18/1.10 = 18/110 = 9/55
PE/FR = 0.36/2.20 = 36/220 = 9/55
∴ PE/EQ = PF/FR.
So, EF is parallel to QR.