To determine how many distinct rectangles can be formed on a 4 × 4 geoboard (a grid with 4 rows and 4 columns), where no two rectangles are congruent, we need to carefully analyze the problem.

Key Points:

1. Geoboard Layout: The 4 × 4 grid consists of 5 horizontal and 5 vertical lines (since the grid has one more line than the number of spaces between the pins).
2. Rectangle Properties: A rectangle on the geoboard is defined by selecting two distinct horizontal lines and two distinct vertical lines. The points where these lines intersect form the corners of the rectangle.

Steps to Calculate:

1. Choose Two Horizontal Lines: There are 5 horizontal lines (since the grid has 4 rows). We need to select 2 distinct lines from these 5. The number of ways to do this is given by the combination formula $\binom{5}{2}$. $\binom{5}{2} = \frac{5 \times 4}{2} = 10$

2. Choose Two Vertical Lines: Similarly, there are 5 vertical lines (since the grid has 4 columns). We need to select 2 distinct lines from these 5. The number of ways to do this is also $\binom{5}{2}$, which is 10.

3. Total Number of Rectangles: The total number of rectangles that can be formed is the product of the number of ways to choose two horizontal lines and the number of ways to choose two vertical lines: $10 \times 10 = 100$

Removing Congruent Rectangles:

Now, we need to ensure that we are counting only non-congruent rectangles. Rectangles are congruent if they have the same dimensions (length and width). This depends on the relative positions of the chosen horizontal and vertical lines.

• The length of a rectangle is the horizontal distance between the selected vertical lines.
• The width of a rectangle is the vertical distance between the selected horizontal lines.

Each distinct combination of horizontal and vertical line selections will lead to a unique rectangle, but there will be congruency when the dimensions (length and width) are identical. For instance, a rectangle with a 2-unit length and a 1-unit width is congruent to any other rectangle with the same dimensions.

To simplify, we need to consider only the distinct pairs of lengths and widths formed by the selections of horizontal and vertical lines. These distinct pairs correspond to unique sets of dimensions (width × length), and counting them requires considering how many different combinations of lengths and widths are possible.

Final Count:

After accounting for congruence, we conclude that there are 16 non-congruent rectangles. These represent all possible unique combinations of lengths and widths that can be formed by selecting pairs of horizontal and vertical lines.

---

Would you like to see further details on the method used to account for congruent rectangles, or do you have any other questions? Here's a list of related questions you might find interesting:

1. How does this approach change if we used a different size geoboard (e.g., 3 × 3 or 5 × 5)?
2. Can you calculate the number of congruent rectangles and then subtract them from the total?
3. What is the total number of squares that can be formed on a 4 × 4 geoboard?
4. How would you approach counting non-congruent rectangles in a 6 × 6 geoboard?
5. What are the formulas for counting non-congruent rectangles in general for any grid size?

Tip: When counting distinct shapes like rectangles on a grid, always consider the grid's symmetry and how dimensions repeat across different locations!