Difference between revisions of "2022 AMC 8 Problems/Problem 18"

Revision as of 12:17, 28 January 2022

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?

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

~MRENTHUSIASM

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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