2022 AMC 8 Problems/Problem 18

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Problem

The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle?

$\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80$

Solution

The midpoints of the four sides of every rectangle are the vertices of a rhombus whose area is half the area of the rectangle.

Note that $(-3,0), (2,0), (5,4),$ and $(0,4)$ are the vertices of a square whose diagonal is $5.$ The area of this square is $\frac{5\cdot5}{2}=\frac{25}{2},$ so the area of the rectangle is $\frac{25}{2}\cdot2=\boxed{\textbf{(B) } 25}.$

~MRENTHUSIASM

See Also

2022 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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All AJHSME/AMC 8 Problems and Solutions

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