Question

M and N are points on the sides PQ and PR respectively of a $△$ PQR. For each of the following cases, state whether MN $\parallel$ QR :

(i) PM = 4 cm,QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm

(ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm

Answer 1

Here, using the corollary of basic proportionally theorem which states that if a line passing through the two sides of the triangle cuts it proportionally, then the line is parallel to the third side. So,

$\left(i\right)$ $\frac{PM}{QM}=\frac{4}{4.5}=\frac{8}{9}$

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

$\therefore$ $\frac{PM}{QM}=\frac{PN}{NR}$

Thus, as $MN$ cuts the sides $PQ$ and $PR$ proportionally, so $MN\parallel QR$.

$\therefore$ $MN\parallel QR$

$\left(ii\right)$ $\frac{PM}{QM}=\frac{0.16}{1.28-0.16}=\frac{0.16}{1.12}=\frac{1}{7}$

$\frac{PN}{NR}=\frac{0.32}{2.56-0.32}=\frac{0.32}{2.24}=\frac{1}{7}$

$\therefore$ $\frac{PM}{QM}=\frac{PN}{NR}$

Thus, as $MN$ cuts the sides $PQ$ and $PR$ proportionally, so $MN\parallel QR$.

$\therefore$ $MN\parallel QR$