In the figure, $PQRS$ is a square and $T$ and $U$ are, respectively, the mid-points of $PS$ and $QR$. Find the area of $\triangle OTS$ if $PQ = 8\ cm$.
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Given:

$PQRS$ is a square and $T$ and $U$ are, respectively, the mid-points of $PS$ and $QR$.

To do:

We have to find the area of $\triangle OTS$ if $PQ = 8\ cm$.

Solution:

$T$ and $U$ are the mid-points of sides $PS$ and $QR$.

This implies,

$TU \parallel PQ, TO \parallel PQ$

In $\triangle RQS$,

$T$ is the mid-point of $PS$ and $TO \parallel PQ$

$O$ is the mid point of $SQ$

$\mathrm{TO}=\frac{1}{2} \mathrm{PQ}$

$=\frac{1}{2} \times 8$

$=4 \mathrm{~cm}$

$\mathrm{TS}=\frac{1}{2} \times \mathrm{PS}$

$=\frac{1}{2} \times 8$

$=4 \mathrm{~cm}$

Area of $\Delta \mathrm{OTS}=\frac{1}{2} \mathrm{OT} \times \mathrm{TS}$

$=\frac{1}{2} \times 4 \times 4$

$=8 \mathrm{~cm}^{2}$

The area of $\triangle OTS$ is $8\ cm^2$.