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(Parallelograms, More Detailed))
m (→‎Solution)
Line 6: Line 6:
 
<math>\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80</math>
 
<math>\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80</math>
  
−
==Solution==
+
==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.

Revision as of 10:18, 29 January 2022

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 $A=(-3,0), B=(2,0), C=(5,4),$ and $D=(0,4)$ are the vertices of a rhombus whose diagonals have lengths $AC=\sqrt{80}$ and $BD=\sqrt{20}.$ It follows that the area of rhombus $ABCD$ is $\frac{\sqrt{80}\cdot\sqrt{20}}{2}=20,$ so the area of the rectangle is $20\cdot2=\boxed{\textbf{(C) } 40}.$

~MRENTHUSIASM

Solution 2(Parallelograms, More Detailed)

Note that if a rectangle has area $A$, the area of the quadrilateral formed by its midpoints is $\frac{A}{2}$. Since $A, B, C, D$ are the midpoints of the rectangle, its area would be $2\cdot[ABCD]$. Now, note that $ABCD$ is a parallelogram since $AB=CD$ and $AB||CD$. Note that the parallelogram's altitude from $D$ to $AB$ is $4$ and $AB=5$. Thus, its area is $4\times 5=20$. The area of the rectangle follows as $20\cdot2=\boxed{\textbf{(C) } 40}.$

~Fruitz

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=170698"