Question

E and F are points on the sides PQ and PR respectively of a $\Delta PQR$. For the following case, state whether EF || QR.
(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm.
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

Answer 1

In $\Delta PQR$, E and F are two points on the sides PQ and PR respectively.
(i) PE = 3.9 cm, EQ = 3 cm (Given)
PF = 3.6 cm, FR = 2.4 cm (Given)
$\therefore \frac{PE}{EQ}=\frac{3.9}{3}=\frac{39}{30}=\frac{13}{10}=1.3$ [By using Basic proportionality theorem]
And, $\frac{PF}{FR}=\frac{3.6}{2.4}=\frac{36}{24}=\frac{3}{2}=1.5$
So, $\frac{PE}{EQ}\ne \frac{PF}{FR}$
Hence, EF is not parallel to QR.

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8cm, RF = 9cm
$\therefore \frac{PE}{QE}=\frac{4}{4.5}=\frac{40}{45}=\frac{8}{9}$ [By using Basic proportionality theorem]
And,$\frac{PF}{RF}=\frac{8}{9}$
So, $\frac{PE}{QE}=\frac{PF}{RF}$
Hence, EF is parallel to QR.