|
|
Line 1: |
Line 1: |
− | ==Problem==
| + | 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>
| + | 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}}
| |