To solve this, let's find the midpoint, the distance between the two points, and each point’s distance from the origin.

1. Midpoint

The formula for the midpoint between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

For points $(4, 2)$ and $(-3, 4)$:

$\text{Midpoint} = \left( \frac{4 + (-3)}{2}, \frac{2 + 4}{2} \right) = \left( \frac{1}{2}, 3 \right)$

2. Distance Between the Points

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

For points $(4, 2)$ and $(-3, 4)$:

$d = \sqrt{(-3 - 4)^2 + (4 - 2)^2} = \sqrt{(-7)^2 + (2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.28$

3. Distance of Each Point from the Origin

The distance from the origin $(0, 0)$ to a point $(x, y)$ is given by:

$d = \sqrt{x^2 + y^2}$

For $(4, 2)$:

$d = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47$

For $(-3, 4)$:

$d = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Summary

• Midpoint: $\left( \frac{1}{2}, 3 \right)$
• Distance between points: $\sqrt{53} \approx 7.28$
• Distance from origin:
  • For $(4, 2)$: $\sqrt{20} \approx 4.47$
  • For $(-3, 4)$: $5$

Would you like more details or have any questions?

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1. How to find the slope between two points?
2. What if one point is at the origin?
3. How to find the equation of a line passing through two points?
4. What are the coordinates of points equidistant from the origin?
5. How does the distance formula relate to the Pythagorean theorem?

Tip: When calculating distances, ensure all signs are handled correctly for each coordinate.