AMC8 2022

AMC8 2022 · Q18

AMC8 2022 · Q18. It mainly tests Area & perimeter, Coordinate geometry.

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?

一个矩形的四边中点是$(-3,0), (2,0), (5,4)$和$(0,4)$。这个矩形的面积是多少？

(A) 20 20

(B) 25 25

(D) 50 50

(E) 80 80

Answer

Correct choice: (C)

正确答案：（C）

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 the diagonals of the rhombus have the same lengths as the sides of the rectangle. Let $A=(-3,0), B=(2,0), C=(5,4),$ and $D=(0,4).$ Note that $A,B,C,$ and $D$ are the vertices of a rhombus whose diagonals have lengths $AC=4\sqrt{5}$ and $BD=2\sqrt{5}.$ It follows that the dimensions of the rectangle are $4\sqrt{5}$ and $2\sqrt{5},$ so the area of the rectangle is $4\sqrt{5}\cdot2\sqrt{5}=\boxed{\textbf{(C) } 40}.$

每个矩形的四边中点构成一个菱形的顶点，其面积是矩形面积的一半：注意菱形的对角线长度与矩形的边长相同。设$A=(-3,0), B=(2,0), C=(5,4), D=(0,4)$。注意$A,B,C,D$是一个菱形的顶点，其对角线长度为$AC=4\sqrt{5}$和$BD=2\sqrt{5}$。由此，矩形的边长为$4\sqrt{5}$和$2\sqrt{5}$，所以矩形的面积是$4\sqrt{5}\cdot2\sqrt{5}=\boxed{\textbf{(C) } 40}$。

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