Difference between revisions of "2022 AMC 8 Problems/Problem 18"
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Define points <math>A,B,C,</math> and <math>D</math> as Solution 1 does. Since <math>A,B,C,</math> and <math>D</math> are the midpoints of the rectangle, the rectangle's area is <math>2[ABCD].</math> Now, note that <math>ABCD</math> is a parallelogram since <math>AB=CD</math> and <math>\overline{AB}\parallel\overline{CD}.</math> As the parallelogram's height from <math>D</math> to <math>\overline{AB}</math> is <math>4</math> and <math>AB=5,</math> its area is <math>4\cdot5=20.</math> Therefore, the area of the rectangle is <math>20\cdot2=\boxed{\textbf{(C) } 40}.</math> | Define points <math>A,B,C,</math> and <math>D</math> as Solution 1 does. Since <math>A,B,C,</math> and <math>D</math> are the midpoints of the rectangle, the rectangle's area is <math>2[ABCD].</math> Now, note that <math>ABCD</math> is a parallelogram since <math>AB=CD</math> and <math>\overline{AB}\parallel\overline{CD}.</math> As the parallelogram's height from <math>D</math> to <math>\overline{AB}</math> is <math>4</math> and <math>AB=5,</math> its area is <math>4\cdot5=20.</math> Therefore, the area of the rectangle is <math>20\cdot2=\boxed{\textbf{(C) } 40}.</math> | ||
| + | ~Fruitz | ||
| + | |||
| + | ==Video Solution by Math-X (First understand the problem!!!)== | ||
| + | https://youtu.be/oUEa7AjMF2A?si=8oNmauhAnW5T5vEX&t=3182 | ||
| + | |||
| + | ~Math-X | ||
| − | |||
==Video Solution (🚀Just 2 min!🚀)== | ==Video Solution (🚀Just 2 min!🚀)== | ||
https://youtu.be/5Vti6QS7TfU | https://youtu.be/5Vti6QS7TfU | ||
Latest revision as of 13:53, 23 November 2023
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 ![$2[ABCD].$](http://latex.artofproblemsolving.com/1/e/2/1e2ceebf98aecc6c1a157ef5aeeeacf3ad98408d.png)
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 by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=8oNmauhAnW5T5vEX&t=3182
~Math-X
Video Solution (🚀Just 2 min!🚀)
~Education, the Study of Everything
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
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
