midpoint $$ _{\mathrm{BD}}=\left(\frac{-2+x}{2} ; \frac{4+y}{2}\right) $$

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The expression you've provided represents the **midpoint** of a line segment $$ \overline{\mathrm{BD}} $$ in a Cartesian coordinate system. Let's break it down and understand how to work with it.

### Midpoint Formula

The midpoint $$ M $$ of a line segment connecting two points $$ B(x_1, y_1) $$ and $$ D(x_2, y_2) $$ is given by the formula:

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

### Your Expression

You've provided the midpoint expression as:

$$
_{\mathrm{BD}} = \left( \frac{-2 + x}{2} , \frac{4 + y}{2} \right)
$$

This suggests that one endpoint of the segment $$ \overline{\mathrm{BD}} $$ is known, and you're either solving for the other endpoint or verifying the midpoint.

Let's assume:
- Point $$ B $$ has coordinates $$ (-2, 4) $$.
- Point $$ D $$ has coordinates $$ (x, y) $$.
- The midpoint $$ M $$ has coordinates $$ \left( \frac{-2 + x}{2}, \frac{4 + y}{2} \right) $$.

### Possible Scenarios

1. **Finding the Midpoint Given Endpoints**:
   If you know both endpoints $$ B $$ and $$ D $$, you can directly substitute their coordinates into the midpoint formula to find $$ M $$.

2. **Finding an Endpoint Given the Midpoint and the Other Endpoint**:
   If you know the midpoint $$ M $$ and one endpoint (say $$ B $$), you can solve for the unknown endpoint $$ D(x, y) $$.

For example, if the midpoint $$ M $$ is known to be $$ (m_x, m_y) $$, then:
   $$
   \frac{-2 + x}{2} = m_x \quad \Rightarrow \quad x = 2m_x + 2
   $$
   $$
   \frac{4 + y}{2} = m_y \quad \Rightarrow \quad y = 2m_y - 4
   $$

3. **Verifying if a Point is the Midpoint**:
   If you have a candidate point for the midpoint, you can substitute it into the formula to verify its correctness based on known endpoints.

### Example

**Problem**: Given that point $$ B $$ is $$ (-2, 4) $$ and the midpoint $$ M $$ of $$ \overline{\mathrm{BD}} $$ is $$ (3, 1) $$, find the coordinates of point $$ D $$.

**Solution**:
Using the midpoint formula:
$$
\left( \frac{-2 + x}{2}, \frac{4 + y}{2} \right) = (3, 1)
$$

Set up the equations:
$$
\frac{-2 + x}{2} = 3 \quad \Rightarrow \quad -2 + x = 6 \quad \Rightarrow \quad x = 8
$$
$$
\frac{4 + y}{2} = 1 \quad \Rightarrow \quad 4 + y = 2 \quad \Rightarrow \quad y = -2
$$

**Answer**: Point $$ D $$ is $$ (8, -2) $$.

### Conclusion

If you provide more context or specify what you need assistance with regarding the midpoint expression, I can offer a more tailored solution. Whether it's solving for unknown coordinates, verifying a midpoint, or applying the formula in a particular problem, feel free to share additional details!
The midpoint $$ M $$ of segment $$ \overline{\mathrm{BD}} $$ is $$ \left( \frac{-2 + x}{2}, \frac{4 + y}{2} \right) $$.

midpoint

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