Question

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$. For each of the following cases state whether $EF\parallel QR$
(1) $PE=3.9cm,EQ=3cm,PF=3.6cmandFR=2.4$
(2) $PE=4cm,QE=4.5cm,PF=8cmandRF=9cm$
(3) $PQ=1.28cm\mathrm{PR}=2.56cm,PE=0.18cmandPF=0.36cm$

Answer 1

By basic proportionality theorm ,

if $EF||QR$ then $\frac{PE}{EQ}=\frac{PF}{FR}$

(1) $PE=3.9cm,EQ=3cm,PF=3.6cmandFR=2.4$

$\frac{3.9}{3}=\frac{3.6}{2.4}\frac{1.3}{1}\ne \frac{3}{2}$

So EF is not parallel to QR

(2) $PE=4cm,QE=4.5cm,PF=8cmandRF=9cm$

$\frac{4}{4.5}=\frac{8}{9}\frac{8}{9}=\frac{8}{9}$

So EF is parallel to QR

(3) $PQ=1.28cm\mathrm{PR}=2.56cm,PE=0.18cmandPF=0.36cm$

$\frac{1.28}{2.56}=\frac{0.18}{0.36}\frac{1}{2}=\frac{1}{2}$

So EF is parallel to QR