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Find the number of ways of choosing three points from the grid below, so that they form an isosceles right triangle.

 Feb 11, 2023
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The number of ways of choosing three points from the grid to form an isosceles right triangle is the same as the number of lattice points that are the vertices of an isosceles right triangle.

We consider the four cases for the right angle of the isosceles right triangle:

The right angle is at the origin (0, 0). In this case, the other two points are at (a, 0) and (0, a), where a is a positive integer. There are a total of \(sqrt(2n) - 1\) points of this type for n = 50.

The right angle is at the point (a, a), where a is a positive integer. In this case, the other two points are at (a, 0) and (0, a). There are a total of \(sqrt(2n) - 1\) points of this type for n = 50.

The right angle is at the point (-a, a), where a is a positive integer. In this case, the other two points are at (0, a) and (-a, 0). There are a total of \(sqrt(2n) - 1\) points of this type for n = 50.

The right angle is at the point (-a, -a), where a is a positive integer. In this case, the other two points are at (0, -a) and (-a, 0). There are a total of \(sqrt(2n) - 1\)  points of this type for n = 50.

Thus, the total number of isosceles right triangles is \(4 * (sqrt(2n) - 1) = 4 * (sqrt(100) - 1) = 4 * 10 - 4 = 36\).

Therefore, there are 36 different ways of choosing three points from the grid to form an isosceles right triangle.

 Feb 11, 2023

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