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 the diagonals of the rhombus have the same lengths as the sides of the rectangle.
Let and Note that 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
~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
Video Solution by Ismail.maths
https://www.youtube.com/watch?v=JHBcnevL5_U
~Ismail.maths93
Video Solution
https://youtu.be/hs6y4PWnoWg?t=188
~STEMbreezy
Video Solution
~savannahsolver
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AJHSME/AMC 8 Problems and Solutions |
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