Difference between revisions of "2022 AMC 8 Problems/Problem 1"
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==Solution 2== | ==Solution 2== | ||
− | + | There are <math>5</math> lattice points in the interior of the logo and <math>12</math> lattice points on the boundary of the logo. Because of Pick's Theorem, the area of the logo is <math>5+\frac{12}{2}-1=\boxed{\textbf{(A) } 10}</math>. | |
~MathFun1000 | ~MathFun1000 | ||
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− | The area of the big square is <math>4</math>, and the area of each triangle is <math>0.5</math>. There are <math>12</math> of these triangles, so the total area of all the triangles is <math>0.5\cdot12=6</math>. Therefore, the area of the entire figure is <math>4+6=\boxed{\textbf{(A) } 10}</math>. | + | The area of the big square is <math>4</math>, and the area of each triangle is <math>0.5</math>. There are <math>12</math> of these triangles, so the total area of all the triangles is <math>0.5\cdot12=6</math>. Therefore, the area of the entire figure is <math>4+6=\boxed{\textbf{(A) } 10}</math>. |
− | == | + | ~RocketScientist |
+ | |||
+ | ==Solution 5 (Shoelace Theorem)== | ||
The coordinates are <math>(1,2), (2,1), (3,2), (4,1), (5,2), (4,3), (5,4), (4,5), (3,4), (2,5), (1,4), (2,3)</math> | The coordinates are <math>(1,2), (2,1), (3,2), (4,1), (5,2), (4,3), (5,4), (4,5), (3,4), (2,5), (1,4), (2,3)</math> | ||
− | Use the [[Shoelace Theorem]] to get <math>\boxed{\textbf{(A)} ~10}</math> | + | Use the [[Shoelace Theorem]] to get <math>\boxed{\textbf{(A)} ~10}</math>. |
+ | |||
+ | ==Solution 6 (Quick) == | ||
+ | If the triangles are rearranged such that the gaps are filled, there would be a <math>4</math> by <math>2</math> rectangle, and two <math>1</math> by <math>1</math> squares are present. Thus, the answer is <math>\boxed{\textbf{(A)} ~10}</math>. | ||
+ | |||
+ | ~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!!!)== |
− | + | https://youtu.be/8TyfWyTOxLI | |
− | + | ~Education, the Study of Everything | |
− | |||
+ | ==Video Solution 3== | ||
+ | https://www.youtube.com/watch?v=Ij9pAy6tQSg | ||
~Interstigation | ~Interstigation | ||
+ | |||
+ | ==Video Solution 4== | ||
+ | https://youtu.be/mKAqJxZMKTM | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution 5 == | ||
+ | https://youtu.be/Q0R6dnIO95Y | ||
+ | |||
+ | ~STEMbreezy | ||
+ | |||
+ | ==Video Solution 6 == | ||
+ | https://www.youtube.com/watch?v=pGpDR0hm6qs | ||
+ | |||
+ | ~harungurcan | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|before=First Problem|num-a=2}} | {{AMC8 box|year=2022|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:22, 21 December 2023
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.