Difference between revisions of "2022 AMC 8 Problems/Problem 18"
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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 <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{ | + | 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{AC\cdot BD}{2}=20,</math> so the area of the rectangle is <math>20\cdot2=\boxed{\textbf{(C) } 40}.</math> |
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 13:38, 28 January 2022
Problem
The midpoints of the four sides of a rectangle are and
What is the
area of the rectangle?
Solution
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 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
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 |
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