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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~peelybonehead | ~peelybonehead | ||
− | ==Video Solution== | + | ==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 | https://www.youtube.com/watch?v=Ij9pAy6tQSg | ||
~Interstigation | ~Interstigation | ||
− | ==Video Solution | + | ==Video Solution 4== |
https://youtu.be/mKAqJxZMKTM | https://youtu.be/mKAqJxZMKTM | ||
~savannahsolver | ~savannahsolver | ||
− | ==Video Solution == | + | ==Video Solution 5 == |
https://youtu.be/Q0R6dnIO95Y | https://youtu.be/Q0R6dnIO95Y | ||
~STEMbreezy | ~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.