Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 8 Problems/Problem 18"
(→‎Solution 2)
 
(39 intermediate revisions by 13 users not shown)
Line 8: Line 8:
 
==Solution 1==
 
==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.
+
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.
  
−
Note that <math>A=(-3,0), B=(2,0), C=(5,4),</math> and <math>D=(0,4)</math> are the vertices of a rhombus whose diagonals have lengths <math>AC=\sqrt{80}</math> and <math>BD=\sqrt{20}.</math> It follows that the area of rhombus <math>ABCD</math> is <math>\frac{\sqrt{80}\cdot\sqrt{20}}{2}=20,</math> so the area of the rectangle is <math>20\cdot2=\boxed{\textbf{(C) } 40}.</math>
+
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
 
~MRENTHUSIASM
Line 16: Line 16:
 
==Solution 2==
 
==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>.
+
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, D</math> are the midpoints of the rectangle, its area is <math>2\cdot[ABCD]</math>. Now, note that <math>ABCD</math> is a parallelogram since <math>AB=CD</math> and <math>AB\parallel CD</math>. As the parallelogram's altitude from <math>D</math> to <math>AB</math> is <math>4</math> and <math>AB=5</math>, its area is <math>4\times 5=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
 
~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
  
 
==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 01:46, 4 November 2024

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?

$\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80$

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 $A=(-3,0), B=(2,0), C=(5,4),$ and $D=(0,4).$ Note that $A,B,C,$ and $D$ are the vertices of a rhombus whose diagonals have lengths $AC=4\sqrt{5}$ and $BD=2\sqrt{5}.$ It follows that the dimensions of the rectangle are $4\sqrt{5}$ and $2\sqrt{5},$ so the area of the rectangle is $4\sqrt{5}\cdot2\sqrt{5}=\boxed{\textbf{(C) } 40}.$

~MRENTHUSIASM

Solution 2

If a rectangle has area $K,$ then the area of the quadrilateral formed by its midpoints is $\frac{K}{2}.$

Define points $A,B,C,$ and $D$ as Solution 1 does. Since $A,B,C,$ and $D$ are the midpoints of the rectangle, the rectangle's area is $2[ABCD].$ Now, note that $ABCD$ is a parallelogram since $AB=CD$ and $\overline{AB}\parallel\overline{CD}.$ As the parallelogram's height from $D$ to $\overline{AB}$ is $4$ and $AB=5,$ its area is $4\cdot5=20.$ Therefore, the area of the rectangle is $20\cdot2=\boxed{\textbf{(C) } 40}.$

~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

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_8_Problems/Problem_18&oldid=231523"