How many rectangles have four vertices on the grid? Note that the sides of a rectangle must be parallel to the sides of the grid.

6×4 grid

To determine how many rectangles can be formed on a \(6 \times 4\) grid where the sides of the rectangles are parallel to the sides of the grid, we need to follow these steps:

### Step 1: Understand the grid
- A \(6 \times 4\) grid has 7 vertical lines and 5 horizontal lines, because a \(n \times m\) grid has \(n+1\) vertical and \(m+1\) horizontal lines.
- To form a rectangle, we need to choose two distinct vertical lines and two distinct horizontal lines. 

### Step 2: Calculate the number of ways to choose lines
- **Choosing two vertical lines**: There are 7 vertical lines, and we need to choose 2. The number of ways to choose 2 vertical lines from 7 is given by the combination:
  \[
  \binom{7}{2} = \frac{7 \times 6}{2} = 21
  \]
- **Choosing two horizontal lines**: Similarly, there are 5 horizontal lines, and we need to choose 2. The number of ways to choose 2 horizontal lines from 5 is:
  \[
  \binom{5}{2} = \frac{5 \times 4}{2} = 10
  \]

### Step 3: Multiply the results
- The total number of rectangles is the product of these two combinations:
  \[
  \text{Total rectangles} = \binom{7}{2} \times \binom{5}{2} = 21 \times 10 = 210
  \]

Thus, the total number of rectangles that can be formed on a \(6 \times 4\) grid is **210**.

Would you like more details on any part of this solution?

Here are some related questions to explore:
1. How many squares can be formed on a \(6 \times 4\) grid?
2. What if the grid was \(8 \times 5\)? How many rectangles would that grid have?
3. How many ways can you choose two lines from a set of 10 lines?
4. How do we derive the formula for combinations?
5. What are the properties of a grid that affect the number of possible rectangles?

**Tip**: To form a square, both the vertical and horizontal distances must be equal, so you need to focus on grids where this is possible!

How many rectangles have four vertices on the grid? Note that the sides of a rectangle must be parallel to the sides of the grid. 6×4 grid

Explore how to calculate the number of rectangles on a 6×4 grid where sides are parallel to the grid lines. This detailed solution uses combinatorics to determine that there are 210 possible rectangles, covering key geometry and combinatorial concepts. Suitable for students in Grades 8-10.

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