In Delta XYZ,P and Q are the midpoints of ove | vedclass

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(A) According to the Midpoint Theorem,if $P$ and $Q$ are the midpoints of sides $\overline{XY}$ and $\overline{XZ}$ of $\Delta XYZ$,then $\overline{PQ

Vedclass · Accepted Answer

$35$

Vedclass · Answer

(A) According to the Midpoint Theorem,if $P$ and $Q$ are the midpoints of sides $\overline{XY}$ and $\overline{XZ}$ of $\Delta XYZ$,then $\overline{PQ} \parallel \overline{YZ}$ and $PQ = \frac{1}{2} YZ$.Since $\overline{PQ} \parallel \overline{YZ}$,$\Delta XPQ$ is similar to $\Delta XYZ$ by the $AA$ similarity criterion ($\angle X = \angle X$ and $\angle XPQ = \angle XYZ$ as corresponding angles).The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.Therefore,$\frac{	ext{Area}(\Delta XPQ)}{	ext{Area}(\Delta XYZ)} = \left( \frac{PQ}{YZ} ight)^2$.Substituting the ratio $\frac{PQ}{YZ} = \frac{1}{2}$,we get $\frac{	ext{Area}(\Delta XPQ)}{140} = \left( \frac{1}{2} ight)^2 = \frac{1}{4}$.Thus,$	ext{Area}(\Delta XPQ) = \frac{140}{4} = 35$.

In $\Delta XYZ$,$P$ and $Q$ are the midpoints of $\overline{XY}$ and $\overline{XZ}$ respectively. If the area of $\Delta XYZ$ is $140$,find the area of $\Delta XPQ$.

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