Difference between revisions of "2022 AMC 8 Problems/Problem 18"

m (Problem== The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle? $\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80$)
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==Problem==
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The rizzler of the ohio is being fanum taxed by the gyaat maximus, who has 1 billion aura, and is a gooner
  
The midpoints of the four sides of a rectangle are <math>(-3,0), (2,0), (5,4),</math> and <math>(0,4).</math> What is the
 
area of the rectangle?
 
  
<math>\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80</math>
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THE GYAAAT
 
 
==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 <math>A=(-3,0), B=(2,0), C=(5,4),</math> and <math>D=(0,4).</math> Note that <math>A,B,C,</math> and <math>D</math> are the vertices of a rhombus whose diagonals have lengths <math>AC=4\sqrt{5}</math> and <math>BD=2\sqrt{5}.</math> It follows that the dimensions of the rectangle are <math>4\sqrt{5}</math> and <math>2\sqrt{5},</math> so the area of the rectangle is <math>4\sqrt{5}\cdot2\sqrt{5}=\boxed{\textbf{(C) } 40}.</math>
 
 
 
~MRENTHUSIASM
 
 
 
==Solution 2==
 
 
 
If a rectangle has area <math>K,</math> then the area of the quadrilateral formed by its midpoints is <math>\frac{K}{2}.</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!🚀)==
 
https://youtu.be/5Vti6QS7TfU
 
 
 
~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==
 
https://youtu.be/9-TlEV5SGqM
 
 
 
~savannahsolver
 
 
 
==Video Solution by Dr. David==
 
https://youtu.be/wiHoXLBr8fQ
 
 
 
==See Also==
 
{{AMC8 box|year=2022|num-b=17|num-a=19}}
 
{{MAA Notice}}
 

Revision as of 22:09, 28 December 2024

The rizzler of the ohio is being fanum taxed by the gyaat maximus, who has 1 billion aura, and is a gooner


THE GYAAAT