2019 AMC 8

Complete problem set with solutions and individual problem pages

Problem 6 Easy

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?

  • A.

    \frac15

  • B.

    \frac14

  • C.

    \frac25

  • D.

    \frac{9}{20}

  • E.

    \frac12

Answer:C

Solution 1

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}}.

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