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# $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \| QR$:

$PE = 4\ cm, QE = 4.5\ cm, PF = 8\ cm$ and $RF = 9\ cm$

Given:

$PE = 4\ cm, QE = 4.5\ cm, PF = 8\ cm$ and $RF = 9\ cm$

To do:

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

Solution:

We know that,

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

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

$\frac{PF}{FR}=\frac{8}{9}$

$\frac{PE}{EQ}=\frac{PF}{FR}$

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

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