Let's solve both problems step-by-step!

1. Distance Between Two Points

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

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

Here, the points are:

• $(-2, 1)$ as $(x_1, y_1)$
• $(4, 3)$ as $(x_2, y_2)$

Substituting the values:

$d = \sqrt{(4 - (-2))^2 + (3 - 1)^2}$ $d = \sqrt{(4 + 2)^2 + (2)^2}$ $d = \sqrt{6^2 + 2^2}$ $d = \sqrt{36 + 4}$ $d = \sqrt{40}$

Now, approximate the square root: $\sqrt{40} \approx 6.3$

Final Answer:

The distance between the points is 6.3.

2. Midpoint and Quadrant

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

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

Substitute the points $(-2, 1)$ and $(4, 3)$:

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

Quadrant:

The midpoint $(1, 2)$ is in Quadrant I because both $x > 0$ and $y > 0$.

Final Answers:

1. Distance = $6.3$
2. Midpoint = $(1, 2)$
3. Quadrant = Quadrant I

Do you want me to explain any step further? Let me know!

Related Questions:

1. How do you calculate the distance between two points in 3D space?
2. How do you derive the midpoint formula?
3. What are the characteristics of points in different quadrants?
4. How can the Pythagorean Theorem relate to the distance formula?
5. What is the general form of the equation for a line given two points?

Tip: To find the distance between points, always square the differences first before summing them!