find the mid point, distance, from the two points (4,2) and (-3,4) as well as their distance from the origin

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.

Find the midpoint, distance from two points (4,2) and (-3,4), as well as their distance from the origin.

Learn to find the midpoint and distance between two points, (4,2) and (-3,4), along with their distances from the origin. This problem demonstrates coordinate geometry concepts and uses formulas for midpoint and distance calculation, suitable for students in Grades 8-10.

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