In the given figure, points ^@ M ^@ and ^@ N ^@ divide the side | Math Question and Answer

Answer:

Step by Step Explanation:

Through $A$ , let us draw $XAY || BC$ .

Now, $XY || MP || NQ$ are cut by the transversal $AB$ at $A, M,$ and $N$ respectively such that $AM = MN$ .

Also, $XY || MP || NQ$ are cut by the transversal $AC$ at $A, P,$ and $Q$ respectively. $\begin{aligned} \therefore \space\space AP = PQ &&\ldots\text{(i)} &&[\text{ By intercept theorem }] \end{aligned}$
Again, $MP || NQ || BC$ are cut by the transversal $AB$ at $M, N,$ and $B$ respectively such that $MN = NB$ .

Also, $MP || NQ || BC$ are cut by the transversal $AC$ at $P, Q,$ and $C$ respectively. $\begin{aligned} \therefore \space\space PQ = QC &&\ldots\text{(ii)} &&[\text{ By intercept theorem }] \end{aligned}$ Thus, from $eq \space (\text{i})$ and $eq \space (\text{ii})$ , we get $AP = PQ = QC.$

Therefore, $P$ and $Q$ divide $AC$ into three equal parts.

Hence, $AC = 3AP = 3PQ = 3QC$ .

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