In Delta PQR,M and N are the midpoints of ove | vedclass

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(D) In $\Delta PQR$,$M$ and $N$ are the midpoints of sides $\overline{PQ}$ and $\overline{PR}$ respectively.According to the Midpoint Theorem,the line

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$96$

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(D) In $\Delta PQR$,$M$ and $N$ are the midpoints of sides $\overline{PQ}$ and $\overline{PR}$ respectively.According to the Midpoint Theorem,the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.Therefore,$MN \parallel QR$ and $MN = \frac{1}{2} QR$.Since $MN \parallel QR$,$\Delta PMN$ is similar to $\Delta PQR$ by the $AA$ similarity criterion ($\angle PMN = \angle PQR$ and $\angle PNM = \angle PRQ$ as corresponding angles).The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.$\frac{	ext{Area}(\Delta PMN)}{	ext{Area}(\Delta PQR)} = \left( \frac{MN}{QR} ight)^2$.Substituting $MN = \frac{1}{2} QR$,we get $\frac{	ext{Area}(\Delta PMN)}{	ext{Area}(\Delta PQR)} = \left( \frac{1}{2} ight)^2 = \frac{1}{4}$.Given $	ext{Area}(\Delta PMN) = 24$,we have $\frac{24}{	ext{Area}(\Delta PQR)} = \frac{1}{4}$.Therefore,$	ext{Area}(\Delta PQR) = 24 	imes 4 = 96$.

In $\Delta PQR$,$M$ and $N$ are the midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively. If the area of $\Delta PMN$ is $24$,find the area of $\Delta PQR$.

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