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I need help with a counting problem.

 

In how many ways can three points be selected from the  grid below so that they form a right isosceles triangle?

 

 Mar 13, 2023
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To form a right isosceles triangle, we need to select three points such that two of them lie on one of the grid's horizontal lines and the third lies on a line that is perpendicular to that line and passes through the midpoint of the first two points.

There are 5 horizontal lines and 5 vertical lines we can choose from. Let's consider the cases where the two points are on the first horizontal line:

- If we choose the two points that are closest to each other, they are one unit apart, and there are two points on the second horizontal line that would form a right isosceles triangle with them.
- If we choose two points that are two units apart, there are no points on the second horizontal line that would form a right isosceles triangle with them.
- If we choose two points that are three units apart, there is only one point on the second horizontal line that would form a right isosceles triangle with them.

Therefore, for each horizontal line, we have a total of 2 + 0 + 1 = 3 ways to select two points that are one, two, or three units apart. For each of these pairs of points, there is exactly one point on a vertical line that would form a right isosceles triangle with them. Therefore, the total number of ways to select three points that form a right isosceles triangle is:

5 horizontal lines x 3 ways to choose the pair of points on each line x 1 way to choose the third point on a perpendicular line = 15.

Therefore, there are 15 ways to select three points from the given grid that form a right isosceles triangle.

 Mar 13, 2023

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