2022 AMC 8 Problems/Problem 18
Contents
Problem
The midpoints of the four sides of a rectangle are and What is the area of the rectangle?
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 and are the vertices of a rhombus whose diagonals have lengths and It follows that the area of rhombus is so the area of the rectangle is
~MRENTHUSIASM
Solution 2
If a rectangle has area , then the area of the quadrilateral formed by its midpoints is .
Define points and as Solution 1 does. Since are the midpoints of the rectangle, its area is . Now, note that is a parallelogram since and . As the parallelogram's altitude from to is and , its area is . Therefore, the area of the rectangle is .
~Fruitz
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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