Difference between revisions of "2022 AMC 8 Problems/Problem 18"
MRENTHUSIASM (talk | contribs) m (→Solution 1) |
(→Solution 1) |
||
Line 13: | Line 13: | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | Alternatively, we can find the area of the rhombus by utilizing the [[Shoelace Theorem]]. We then get <math>ABCD=\frac{|-12-28|}{2}=20</math>. Subsequently, the area of the rectangle is <math>20\cdot2=\boxed{\textbf{(C) } 40}.</math> | ||
==Solution 2== | ==Solution 2== |
Revision as of 12:21, 7 February 2022
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.
Let and Note that and are the vertices of a rhombus whose diagonals have lengths and It follows that the area of rhombus is so the area of the rectangle is
~MRENTHUSIASM
Alternatively, we can find the area of the rhombus by utilizing the Shoelace Theorem. We then get . Subsequently, the area of the rectangle is
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 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
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.