Difference between revisions of "2019 AMC 8 Problems/Problem 6"

Latest revision as of 20:17, 5 June 2025

Problem

There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?

$[asy] draw((0,0)--(0,8)); draw((0,8)--(8,8)); draw((8,8)--(8,0)); draw((8,0)--(0,0)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((0,6)); dot((0,7)); dot((0,8)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((1,6)); dot((1,7)); dot((1,8)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((2,6)); dot((2,7)); dot((2,8)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((3,6)); dot((3,7)); dot((3,8)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); dot((4,6)); dot((4,7)); dot((4,8)); dot((5,0)); dot((5,1)); dot((5,2)); dot((5,3)); dot((5,4)); dot((5,5)); dot((5,6)); dot((5,7)); dot((5,8)); dot((6,0)); dot((6,1)); dot((6,2)); dot((6,3)); dot((6,4)); dot((6,5)); dot((6,6)); dot((6,7)); dot((6,8)); dot((7,0)); dot((7,1)); dot((7,2)); dot((7,3)); dot((7,4)); dot((7,5)); dot((7,6)); dot((7,7)); dot((7,8)); dot((8,0)); dot((8,1)); dot((8,2)); dot((8,3)); dot((8,4)); dot((8,5)); dot((8,6)); dot((8,7)); dot((8,8)); label("P",(4,4),NE); [/asy]$

$\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}$

Solution 1

$[asy] draw((0,0)--(0,8)); draw((0,8)--(8,8)); draw((8,8)--(8,0)); draw((8,0)--(0,0)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((0,6)); dot((0,7)); dot((0,8)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((1,6)); dot((1,7)); dot((1,8)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((2,6)); dot((2,7)); dot((2,8)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((3,6)); dot((3,7)); dot((3,8)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); dot((4,6)); dot((4,7)); dot((4,8)); dot((5,0)); dot((5,1)); dot((5,2)); dot((5,3)); dot((5,4)); dot((5,5)); dot((5,6)); dot((5,7)); dot((5,8)); dot((6,0)); dot((6,1)); dot((6,2)); dot((6,3)); dot((6,4)); dot((6,5)); dot((6,6)); dot((6,7)); dot((6,8)); dot((7,0)); dot((7,1)); dot((7,2)); dot((7,3)); dot((7,4)); dot((7,5)); dot((7,6)); dot((7,7)); dot((7,8)); dot((8,0)); dot((8,1)); dot((8,2)); dot((8,3)); dot((8,4)); dot((8,5)); dot((8,6)); dot((8,7)); dot((8,8)); label("P",(4,4),NE); draw((0,4)--(3,4)); draw((0,8)--(3,5)); draw((8,8)--(5,5)); draw((4,8)--(4,5)); draw((4,0)--(4,3)); draw((0,0)--(3,3)); draw((8,0)--(5,3)); draw((5,4)--(8,4)); [/asy]$ Lines of symmetry go through point $P$ , and there are $8$ directions the lines could go, and there are $4$ dots at each direction. $\frac{4\times8}{80}=\boxed{\textbf{(C)} \frac{2}{5}}$ .

Solution 2

Divide the grid into 4 4x5 quadrants. Each row of 5 points has 1 point on a horizontal/vertical line of symmetry + 1 point on a diagonal line of symmetry: $\boxed{\textbf{(C)} \frac{2}{5}}$ .

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/IgpayYB48C4?si=AdzSEy4Ocrte4gEU&t=1650

~Math-X

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/PizqK-oBLqk

~Education, the Study of Everything

Video Solution

The Learning Royal : https://youtu.be/8njQzoztDGc

Video Solution 2

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=4L95z9DwlhI&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=7

Video Solution 3

https://youtu.be/TAKmC11vitM

~savannahsolver

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

@@ Line 1: / Line 1: @@
-== Problem 6 ==
+==Problem==
 There are <math>81</math> grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point <math>P</math> is in the center of the square. Given that point <math>Q</math> is randomly chosen among the other <math>80</math> points, what is the probability that the line <math>PQ</math> is a line of symmetry for the square?
@@ Line 209: / Line 209: @@
 Lines of symmetry go through point <math>P</math>, and there are <math>8</math> directions the lines could go, and there are <math>4</math> dots at each direction.<math>\frac{4\times8}{80}=\boxed{\textbf{(C)} \frac{2}{5}}</math>.
-==Also See==
+== Solution 2 ==
+Divide the grid into 4 4x5 quadrants. Each row of 5 points has 1 point on a horizontal/vertical line of symmetry + 1 point on a diagonal line of symmetry: <math>\boxed{\textbf{(C)} \frac{2}{5}}</math>.
+==Video Solution by Math-X (First fully understand the problem!!!)==
+https://youtu.be/IgpayYB48C4?si=AdzSEy4Ocrte4gEU&t=1650
+~Math-X
+==Video Solution (HOW TO THINK CREATIVELY!!!)==
+https://youtu.be/PizqK-oBLqk
+~Education, the Study of Everything
+== Video Solution ==
+The Learning Royal : https://youtu.be/8njQzoztDGc
+== Video Solution 2 ==
+Solution detailing how to solve the problem: https://www.youtube.com/watch?v=4L95z9DwlhI&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=7
+==Video Solution 3==
+https://youtu.be/TAKmC11vitM
+~savannahsolver
+==Video Solution by The Power of Logic(1 to 25 Full Solution)==
+https://youtu.be/Xm4ZGND9WoY
+~Hayabusa1
+==See also==
 {{AMC8 box|year=2019|num-b=5|num-a=7}}
 {{MAA Notice}}
+[[Category:Introductory Combinatorics Problems]]
+[[Category:Introductory Probability Problems]]

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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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