2022 AMC 8 Problems/Problem 1
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Solution 5 (Shoelace Theorem)
- 7 Solution 6 (Quick)
- 8 Video Solution 1 by Math-X (First understand the problem!!!)
- 9 Video Solution 2 (HOW TO CREATIVELY THINK!!!)
- 10 Video Solution 3
- 11 Video Solution 4
- 12 Video Solution 5
- 13 Video Solution 6
- 14 See Also
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?
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
There are lattice points in the interior of the logo and lattice points on the boundary of the logo. Because of Pick's Theorem, the area of the logo 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 .
Solution 6 (Quick)
If the triangles are rearranged such that the gaps are filled, there would be a by rectangle, and two by squares are present. Thus, the answer is .
~peelybonehead
Video Solution 1 by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=7nqtNywjcJi2uIf7&t=62
~Math-X
Video Solution 2 (HOW TO CREATIVELY THINK!!!)
~Education, the Study of Everything
Video Solution 3
https://www.youtube.com/watch?v=Ij9pAy6tQSg ~Interstigation
Video Solution 4
~savannahsolver
Video Solution 5
~STEMbreezy
Video Solution 6
https://www.youtube.com/watch?v=pGpDR0hm6qs
~harungurcan
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.