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

Revision as of 12:39, 23 April 2021

Problem 6

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. $\frac{4\times8}{80}=\boxed{\textbf{(C)} \frac{2}{5}}$ .

Video Solution

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

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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Difference between revisions of "2019 AMC 8 Problems/Problem 6"

Revision as of 12:39, 23 April 2021

Contents

Problem 6

Solution 1

Video Solution

See also

@@ Line 1: / Line 1: @@
-Problem 7
+== Problem 6 ==
-Shauna takes 5 tests, each worth a maximum of a 100 points. Her scores on the first three tests
-were 76, 94, and 87. In order to average an 81 on all five tests, what is the lowest score she could
-earn on one of the two tests?
-A(48)   B(52)    C(66)    D(70)    E(74)
-SOLUTION
+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?
-So far, she has scored 76+94+87=257 points on her tests. She needs to have 81*5=405 points in total to achieve an average of 81 on her 5 tests. To find the lowest score, one of the remaining scores must be the highest it can be (100). She needs to score 405-275=148. For the minimum score , 148-100=48. So its A(48).~heeeeeeheeeeeee
+<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>
+<math>\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}</math>
+==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 <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>.
+== Video Solution ==
+Solution detailing how to solve the problem: https://www.youtube.com/watch?v=4L95z9DwlhI&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=7
+==See also==
+{{AMC8 box|year=2019|num-b=5|num-a=7}}
+{{MAA Notice}}