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The problem asks to find the distance between the point \((-2, 3)\) and the midpoint of the segment joining the points \((-2, -2)\) and \((4, 3)\).

### Step 1: Find the midpoint of the segment
The formula for the midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Given the points \((-2, -2)\) and \((4, 3)\), the midpoint is:
\[
M = \left( \frac{-2 + 4}{2}, \frac{-2 + 3}{2} \right) = \left( \frac{2}{2}, \frac{1}{2} \right) = (1, 0.5)
\]

### Step 2: Find the distance between \((-2, 3)\) and \((1, 0.5)\)
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substitute the points \((-2, 3)\) and \((1, 0.5)\):
\[
d = \sqrt{(1 - (-2))^2 + (0.5 - 3)^2} = \sqrt{(1 + 2)^2 + (0.5 - 3)^2} = \sqrt{3^2 + (-2.5)^2}
\]
\[
d = \sqrt{9 + 6.25} = \sqrt{15.25} \approx 3.91
\]

### Final Answer:
The distance between the point \((-2, 3)\) and the midpoint of the segment is approximately **3.91 units**.

Would you like more details or have any other questions? 

Here are five related questions to expand your understanding:
1. How do you derive the distance formula from the Pythagorean Theorem?
2. What is the geometrical significance of the midpoint in a line segment?
3. How does the distance formula change in 3-dimensional space?
4. Can the midpoint formula be generalized to higher dimensions?
5. What are some real-world applications of using the distance and midpoint formulas?

**Tip:** Always label points clearly when solving geometry problems to avoid confusion between coordinates.

Find the distance between (-2, 3) and the midpoint of the segment joining (-2, -2) and (4, 3).

Learn how to calculate the distance between the point (-2, 3) and the midpoint of the segment joining (-2, -2) and (4, 3). This solution uses the midpoint and distance formulas, covering key geometry concepts. Suitable for Grades 9-11.

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