There is a 4 by� 4 grid of points. Each pair of horizontally adjacent or vertically adjacent points are separated by a distance of 1 unit. How many circles of radius 1 pass through exactly 2 of these grid points? Justify your response. (Note: the circles must be in
the plane of the grid.)
To determine the number of circles of radius 1 that pass through exactly 2 of the grid points, we need to count the number of pairs of adjacent grid points on the 4x4 grid.
First, let's consider the horizontal circles. Each row has 3 adjacent pairs of grid points. Since there are 4 rows, we have a total of 4 * 3 = 12 horizontal pairs.
Next, let's consider the vertical circles. Each column also has 3 adjacent pairs of grid points. Since there are 4 columns, we have a total of 4 * 3 = 12 vertical pairs.
To find the total number of circles passing through exactly 2 grid points, we add the horizontal and vertical pairs:
12 horizontal pairs + 12 vertical pairs = 24 pairs
Each pair represents a unique circle passing through 2 grid points. Therefore, there are 24 circles of radius 1 that pass through exactly 2 of these grid points.