AMC 8 2022 Problem 18
AMC8 2022 年第 18 题
坐标几何★★★☆☆
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?
一个长方形的四条边的中点是(-3，0)，(2，0)，(5，4)和(0，4)。问这个长方形的面积是多少？
- A.20
- B.25
- C.40
- D.50
- E.80
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
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 are the vertices of a rhombus whose diagonals have lengths and It follows that the dimensions of the rectangle are and so the area of the rectangle is
If a rectangle has area $K,$ then the area of the quadrilateral formed by its midpoints is . Define points $A,B,C,$ and as does. Since $A,B,C,$ and are the midpoints of the rectangle, the rectangle's area is $2[ABCD].$ Now, note that is a parallelogram since AB=CD and As the parallelogram's height from to is 4 and AB=5, its area is Therefore, the area of the rectangle is
Graph paper is not permitted on the AMC 8. Still, blank scratch paper is provided, and with that, you can draw a makeshift coordinate plane and "graph" some dots. It doesn't have to be very exact, but do make reasonable estimates on the locations of the points. Once you graph the dots and look around, it becomes apparent that the only rectangle that would make sense would be if the length were from (-3,0) to (5,4), and the width would have to be from (0,4) to (2,0). With that in mind, use the distance formula to solve for the length and width. The distance formula is as follows: Simply put the coordinates into the formula to get the distance between the two points. Using the formula, the length is , and the width is . Now, multiply the length and width together to get 40.žen D.
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