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.