In a 5 by 5 dot grid, there are multiple squares that can be formed using the dots as vertices. To count the number of squares, we can follow these steps:
1. Identify the size of the squares. In this case, we need to find all squares with sides that align with the grid lines (horizontal and vertical). So, we can have squares of sizes 1x1, 2x2, 3x3, 4x4, and 5x5.
2. Count the squares of each size:
- For 1x1 squares, there are 5x5 = 25 squares in total.
- For 2x2 squares, there are (5-1)x(5-1) = 4x4 = 16 squares in total.
- For 3x3 squares, there are (5-2)x(5-2) = 3x3 = 9 squares in total.
- For 4x4 squares, there are (5-3)x(5-3) = 2x2 = 4 squares in total.
- For 5x5 squares, there is only 1 square in total.
3. Add up the counts of squares for each size:
25 + 16 + 9 + 4 + 1 = 55
So, there are a total of 55 squares that can be formed in a 5 by 5 dot grid, where the dots must be used for all 4 vertices