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

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!