M and N are points on the sides PQ and PR respectively of a $\triangle PQR$. For each of the following cases, state whether $MN \parallel QR$:
$PM = 4\ cm$, $QM = 4.5\ cm, PN = 4\ cm, NR = 4.5\ cm$
Given:
$PM=4\ cm, QM=4.5\ cm, PN=4\ cm$ and $NR=4.5\ cm$.
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.
Therefore,
$\frac{PM}{QM}=\frac{4}{4.5}=\frac{4\times2}{4.5\times2}=\frac{8}{9}$
$\frac{PN}{NR}=\frac{4}{4.5}=\frac{4\times2}{4.5\times2}=\frac{8}{9}$
$\frac{PM}{QM}=\frac{PN}{NR}$
Hence, by converse of proportionality theorem $MN$ is parallel to $QR$.
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