Difference between revisions of "2022 AMC 8 Problems/Problem 18"
(Tag: Undo) |
Hsnacademy (talk | contribs) (→Video Solution by Pi Academy (🚀Fast and EZ!🚀)) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 22: | Line 22: | ||
~Fruitz | ~Fruitz | ||
| − | ==Video Solution | + | ==Video Solution (A Clever Explanation You’ll Get Instantly)== |
| − | https://youtu.be/ | + | https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=2153 |
| + | ~hsnacademy | ||
| − | + | ==Video Solution by Pi Academy (🚀Fast and EZ!🚀)== | |
| + | https://youtu.be/Bxp0coLo5r4?si=49fvyWZBrsutgpBt | ||
| − | ==Video Solution | + | ~ AMC8 Learning |
| + | |||
| + | ==Video Solution== | ||
https://youtu.be/5Vti6QS7TfU | https://youtu.be/5Vti6QS7TfU | ||
| Line 36: | Line 40: | ||
~Interstigation | ~Interstigation | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
==Video Solution== | ==Video Solution== | ||
| Line 51: | Line 45: | ||
~savannahsolver | ~savannahsolver | ||
| − | |||
| − | |||
| − | |||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=17|num-a=19}} | {{AMC8 box|year=2022|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 00:59, 11 January 2025
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 (A Clever Explanation You’ll Get Instantly)
https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=2153 ~hsnacademy
Video Solution by Pi Academy (🚀Fast and EZ!🚀)
https://youtu.be/Bxp0coLo5r4?si=49fvyWZBrsutgpBt
~ AMC8 Learning
Video Solution
~Education, the Study of Everything
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=1564
~Interstigation
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. 
