2019 AMC 8 Problems/Problem 6

Problem 6

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

Solution 1

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));
red 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);
 (Error making remote request. Unknown error_msg)

Lines of symmetry go through point P, and there are 32 points on the lines of symmetry. 32/80=. ~heeeeeheeeeeee

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.