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Geometry Difficulty 3.8 AMC 10/12 Find the answer Canada

$PQRS$ is a parallelogram with area 40.

If $T$ and $V$ are the midpoints of sides $PS$ and $RS$ respectively, then the area of $PRVT$ is

Pick one

Solution

Diagonal $PR$ divides parallelogram $PQRS$ into two equal areas.

That is, the area of $\triangle PRS$ is one half of the area of parallelogram $PQRS$ , or 20.

In $\triangle PRS$ , we construct median $RT$ .

(A median is a line segment that joins a vertex of a triangle to the midpoint of its opposite side.)

Median $RT$ divides $\triangle PRS$ into two equal areas since its base, $PS$ , is halved while the height remains the same.
That is, the area of $\triangle TRS$ is one half of the area of $\triangle PRS$ , or 10.

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Similarly, we construct median $TV$ in $\triangle TRS$ , as shown.

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Median $TV$ divides $\triangle TRS$ into two equal areas since its base, $SR$ , is halved while the height remains the same.

That is, the area of $\triangle TVS$ is one half of the area of $\triangle TRS$ , or 5.
The area of $PRVT$ is equal to the area of $\triangle TVS$ subtracted from the area of $\triangle PRS$ , or $20-5=15$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.