2022 AMC 8 Problems/Problem 1
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[hide]Problem
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
usepackage("mathptmx"); defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7)); label("$\textbf{Team}$", (2.1,3)--(3.9,3)); filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5); draw((0,0)--(6,0), gray); draw((0,1)--(6,1), gray); draw((0,2)--(6,2), gray); draw((0,3)--(6,3), gray); draw((0,4)--(6,4), gray); draw((0,5)--(6,5), gray); draw((0,6)--(6,6), gray); draw((0,0)--(0,6), gray); draw((1,0)--(1,6), gray); draw((2,0)--(2,6), gray); draw((3,0)--(3,6), gray); draw((4,0)--(4,6), gray); draw((5,0)--(5,6), gray); draw((6,0)--(6,6), gray); (Error making remote request. Unexpected URL sent back)
Solution 1
Draw the following four lines as shown:
We see these lines split the figure into five squares with side length . Thus, the area is .
~pog ~wamofan
Solution 2
We can apply Pick's Theorem: There are lattice points in the interior and lattice points on the boundary of the figure. As a result, the area is .
~MathFun1000
Solution 3
Notice that the area of the figure is equal to the area of the square subtracted by the triangles that are half the area of each square, which is . The total area of the triangles not in the figure is , so the answer is .
~hh99754539
Solution 4
Draw the following four lines as shown:
The area of the big square is , and the area of each triangle is . There are of these triangles, so the total area of all the triangles is . Therefore, the area of the entire figure is . ~RocketScientist
Solution 5 (Shoelace Theorem)
The coordinates are Use the Shoelace Theorem to get
Quick solution
If the triangles are rearanged such that the gaps are filled, a by rectangle and by squares are present. Thus, the answer is ~peelybonehead
Video Solution
~Interstigation
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
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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