Solution for Question 2: How many rectangles are there in the given figure? Discuss how you found them.

The grid in the image contains 5 rows and 5 columns, forming a 5 × 5 grid. To determine the total number of rectangles in the figure, we need to count all possible rectangles formed by selecting pairs of horizontal and vertical lines.

Step 1: Formula for counting rectangles

The formula for the total number of rectangles in an $m \times n$ grid is:

$\text{Total Rectangles} = \binom{m+1}{2} \cdot \binom{n+1}{2}$

Where:

• $m$ is the number of rows.
• $n$ is the number of columns.
• $\binom{k}{2} = \frac{k(k-1)}{2}$ is the number of ways to choose 2 lines from $k$ lines.

Step 2: Apply the formula to the $5 \times 5$ grid

• In the given grid, there are 6 horizontal lines (5 rows + 1 extra line at the bottom).
• Similarly, there are 6 vertical lines (5 columns + 1 extra line on the right).

So: $\binom{6}{2} = \frac{6 \cdot (6-1)}{2} = \frac{6 \cdot 5}{2} = 15$

• The number of ways to choose 2 horizontal lines is 15.
• The number of ways to choose 2 vertical lines is 15.

Now, multiply these two values to find the total number of rectangles: $\text{Total Rectangles} = 15 \cdot 15 = 225$

Step 3: Verify the solution

Each rectangle is formed by picking two horizontal lines and two vertical lines. The $15 \times 15 = 225$ rectangles include all possible sizes of rectangles, from the smallest $1 \times 1$ to the largest $5 \times 5$.

Final Answer:

There are 225 rectangles in the given figure.

This was calculated using the formula for the total number of rectangles in a grid, considering all combinations of horizontal and vertical lines.

Would you like further clarification or assistance?

Related Questions:

1. What is the number of squares in the same grid?
2. How does the formula for counting rectangles differ from counting squares?
3. How would the answer change if the grid were $6 \times 4$ instead of $5 \times 5$?
4. Can you generalize the formula for $m \times n$ grids to other shapes (e.g., triangles)?
5. How would you determine the number of overlapping rectangles in such a grid?

Tip:

When solving combinatorics problems involving grids, always account for the total number of lines (not just rows/columns) to avoid miscounting.