M and N are points on the sides PQ and PR respectively of a $\triangle PQR$. For each of the following cases, state whether $MN \parallel QR$:
$PQ = 1.28\ cm, PR = 2.56\ cm, PM = 0.16\ cm, PN = 0.32\ cm$

Given:

$PQ=1.28\ cm, PR=2.56\ cm, PM=0.16\ cm$ and $PN=0.32\ cm$. 

To do:

We have to find if $MN\parallel QR$.

Solution:

We know that,

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

$QM=PQ-PM=(1.28-0.16)\ cm=1.12\ cm$

$NR=PR-PN=(2.56-0.32)\ cm=2.24\ cm$

Therefore,

$\frac{PM}{QM}=\frac{0.16}{1.12}=\frac{0.16\times100}{1.12\times100}=\frac{16}{112}=\frac{1}{7}$

$\frac{PN}{NR}=\frac{0.32}{2.24}=\frac{0.32\times100}{2.24\times100}=\frac{32}{224}=\frac{1}{7}$

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

Hence, by converse of proportionality theorem $MN$ is parallel to $QR$. 

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Updated on: 10-Oct-2022

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