how to find the the distance between points on a graph such as : (-2,-1) and (8,4)

To find the distance between two points \((-2, -1)\) and \((8, 4)\) on a graph, you can use the **distance formula**. The distance formula is derived from the Pythagorean theorem and is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where:
- \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

### Step-by-Step Solution:

1. **Identify the coordinates**:
   \[
   (x_1, y_1) = (-2, -1)
   \]
   \[
   (x_2, y_2) = (8, 4)
   \]

2. **Plug the coordinates into the distance formula**:
   \[
   d = \sqrt{(8 - (-2))^2 + (4 - (-1))^2}
   \]

3. **Simplify the expressions inside the square root**:
   \[
   d = \sqrt{(8 + 2)^2 + (4 + 1)^2}
   \]
   \[
   d = \sqrt{10^2 + 5^2}
   \]

4. **Calculate the squares**:
   \[
   d = \sqrt{100 + 25}
   \]

5. **Add the results and find the square root**:
   \[
   d = \sqrt{125} \approx 11.18
   \]

So, the distance between the points \((-2, -1)\) and \((8, 4)\) is approximately **11.18 units**.

Would you like more details or have any questions about this process?

Here are some related questions you might consider:
1. How do you derive the distance formula from the Pythagorean theorem?
2. How would the distance formula change in three dimensions?
3. Can the distance formula be used to find the distance between points in different quadrants?
4. How do you calculate the midpoint between two points?
5. How does the distance between points change when one coordinate is kept constant?

**Tip:** Always double-check your calculations, especially when dealing with square roots and arithmetic operations.

Learn how to calculate the distance between two points (-2, -1) and (8, 4) on a graph using the distance formula. This solution explains the key concepts of coordinate geometry and applies the Pythagorean theorem. Suitable for students in Grades 8-10.

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