the ordered pairs are (3,0), (-4,-5), (5,7), (-2,1)

Given the ordered pairs \((3,0)\), \((-4,-5)\), \((5,7)\), and \((-2,1)\), we can analyze and derive various properties and relations between them. Here are a few things we can do with these points:

1. **Plotting the Points**: We can plot these points on a Cartesian plane to visualize their positions.

2. **Distance Between Points**: Calculate the distances between each pair of points using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]

3. **Midpoints**: Find the midpoint of each line segment joining two points using the midpoint formula:
   \[
   M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]

4. **Slope of Line Segments**: Calculate the slope of the line passing through each pair of points using the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

5. **Equation of Lines**: Determine the equation of the line passing through each pair of points using the point-slope form of the equation of a line:
   \[
   y - y_1 = m(x - x_1)
   \]

### Detailed Analysis

1. **Distance Between Points**

   - Between \((3,0)\) and \((-4,-5)\):
     \[
     d = \sqrt{(3 - (-4))^2 + (0 - (-5))^2} = \sqrt{(3 + 4)^2 + (0 + 5)^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}
     \]

   - Between \((3,0)\) and \((5,7)\):
     \[
     d = \sqrt{(5 - 3)^2 + (7 - 0)^2} = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}
     \]

   - Between \((3,0)\) and \((-2,1)\):
     \[
     d = \sqrt{(3 - (-2))^2 + (0 - 1)^2} = \sqrt{(3 + 2)^2 + (0 - 1)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}
     \]

   - And similarly for the other pairs.

2. **Midpoints**

   - Between \((3,0)\) and \((-4,-5)\):
     \[
     M = \left( \frac{3 + (-4)}{2}, \frac{0 + (-5)}{2} \right) = \left( \frac{-1}{2}, \frac{-5}{2} \right)
     \]

   - Between \((3,0)\) and \((5,7)\):
     \[
     M = \left( \frac{3 + 5}{2}, \frac{0 + 7}{2} \right) = \left( 4, \frac{7}{2} \right)
     \]

   - And similarly for the other pairs.

3. **Slopes**

   - Between \((3,0)\) and \((-4,-5)\):
     \[
     m = \frac{-5 - 0}{-4 - 3} = \frac{-5}{-7} = \frac{5}{7}
     \]

   - Between \((3,0)\) and \((5,7)\):
     \[
     m = \frac{7 - 0}{5 - 3} = \frac{7}{2}
     \]

   - And similarly for the other pairs.

4. **Equations of Lines**

   - For the line through \((3,0)\) and \((-4,-5)\):
     \[
     y - 0 = \frac{5}{7}(x - 3)
     \]
     Simplifying this equation to slope-intercept form \(y = mx + b\), you get:
     \[
     y = \frac{5}{7}x - \frac{15}{7}
     \]

   - For the line through \((3,0)\) and \((5,7)\):
     \[
     y - 0 = \frac{7}{2}(x - 3)
     \]
     Simplifying this equation to slope-intercept form \(y = mx + b\), you get:
     \[
     y = \frac{7}{2}x - \frac{21}{2}
     \]

   - And similarly for the other pairs.

Would you like a more detailed analysis on any of these points or additional properties and relations between these pairs?

### Relative Questions

1. What is the distance between the points \((5,7)\) and \((-2,1)\)?
2. Can you calculate the midpoint of the line segment joining \((-4,-5)\) and \((5,7)\)?
3. What is the slope of the line passing through the points \((-4,-5)\) and \((-2,1)\)?
4. Determine the equation of the line passing through \((5,7)\) and \((-2,1)\).
5. How would you find the perpendicular bisector of the line segment joining \((3,0)\) and \((5,7)\)?

### Tip

When plotting points on a Cartesian plane, make sure to label the axes and points clearly to avoid any confusion.

The ordered pairs are (3,0), (-4,-5), (5,7), (-2,1).

Explore how to calculate the distances, midpoints, slopes, and equations of lines for the ordered pairs (3,0), (-4,-5), (5,7), and (-2,1). This problem covers key concepts in coordinate geometry, including the distance formula, midpoint formula, and point-slope form. Suitable for Grades 8-10.

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