2022 AMC 8 Problems/Problem 18
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.
Let and 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 and are the midpoints of the rectangle, the rectangle's area is Now, note that is a parallelogram since and As the parallelogram's height from to is and its area is Therefore, the area of the rectangle is
~Fruitz
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=1564
~Interstigation
https://www.youtube.com/watch?v=JHBcnevL5_U ~Ismail.maths93
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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