Find the midpoint and distance between (7,-3) and (5,2)

To find the **midpoint** and **distance** between the points \( (7, -3) \) and \( (5, 2) \), we use the following formulas:

### Midpoint Formula:
The midpoint \( M \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substitute the coordinates \( (x_1, y_1) = (7, -3) \) and \( (x_2, y_2) = (5, 2) \):
\[
M = \left( \frac{7 + 5}{2}, \frac{-3 + 2}{2} \right) = \left( \frac{12}{2}, \frac{-1}{2} \right) = (6, -0.5)
\]

### Distance Formula:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates \( (x_1, y_1) = (7, -3) \) and \( (x_2, y_2) = (5, 2) \):
\[
d = \sqrt{(5 - 7)^2 + (2 + 3)^2} = \sqrt{(-2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29}
\]

Thus, the distance is \( \sqrt{29} \) or approximately \( 5.39 \).

### Final Answer:
- **Midpoint**: \( (6, -0.5) \)
- **Distance**: \( \sqrt{29} \approx 5.39 \)

Would you like more details or have any questions?

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**Related Questions:**
1. What is the formula for finding the slope between two points?
2. Can you find the equation of the line passing through the points \( (7, -3) \) and \( (5, 2) \)?
3. How do you find the distance between a point and a line?
4. What is the general equation for the circle passing through two points?
5. How can you find the centroid of a triangle with given vertices?

**Tip:** The distance formula is derived from the Pythagorean theorem, which connects the horizontal and vertical distances between two points.

Learn how to find the midpoint and distance between the points (7,-3) and (5,2). This problem demonstrates the use of the midpoint formula and distance formula, essential concepts in coordinate geometry. Suitable for students in Grades 7-9.

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