hich of the following formulas can be used to find the midpoint of a line segment with endpoints \( \left(x_{1}, y_{1}\right) \) and \( \left(x_{2}, y_{2}\right) \) ? lect all that apply. \( M=\left(\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}\right) \) \( M=\left(\frac{x_{2}+x_{1}}{2}, \frac{y_{2}+y_{1}}{2}\right) \) \( M=\left(x_{1}+\frac{x_{2}-x_{1}}{2}, y_{1}+\frac{y_{2}-y_{1}}{2}\right) \) \( M=\left(x_{2}-x_{1}, y_{2}-y_{1}\right) \)

To find the midpoint \( M \) of a line segment with endpoints \( \left(x_{1}, y_{1}\right) \) and \( \left(x_{2}, y_{2}\right) \), the standard midpoint formula is: \[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \] Let's evaluate each of the provided formulas: 1. **\( M = \left(\frac{x_{2} - x_{1}}{2}, \frac{y_{2} - y_{1}}{2}\right) \)** - **Analysis:** This formula calculates the **difference** between the coordinates, divided by 2. This gives the **vector** from \( (x_{1}, y_{1}) \) to the midpoint, not the actual coordinates of the midpoint. - **Conclusion:** **Incorrect** for finding the midpoint's coordinates. 2. **\( M = \left(\frac{x_{2} + x_{1}}{2}, \frac{y_{2} + y_{1}}{2}\right) \)** - **Analysis:** This is the **standard midpoint formula**. - **Conclusion:** **Correct**. 3. **\( M = \left(x_{1} + \frac{x_{2} - x_{1}}{2}, y_{1} + \frac{y_{2} - y_{1}}{2}\right) \)** - **Analysis:** Simplifying each component: \[ x_{1} + \frac{x_{2} - x_{1}}{2} = \frac{2x_{1} + x_{2} - x_{1}}{2} = \frac{x_{1} + x_{2}}{2} \] \[ y_{1} + \frac{y_{2} - y_{1}}{2} = \frac{2y_{1} + y_{2} - y_{1}}{2} = \frac{y_{1} + y_{2}}{2} \] This simplifies to the standard midpoint formula. - **Conclusion:** **Correct**. 4. **\( M = \left(x_{2} - x_{1}, y_{2} - y_{1}\right) \)** - **Analysis:** This represents the **vector** from \( (x_{1}, y_{1}) \) to \( (x_{2}, y_{2}) \), not the midpoint. - **Conclusion:** **Incorrect** for finding the midpoint's coordinates. **Therefore, the formulas that correctly find the midpoint are:** - \( M = \left(\frac{x_{2} + x_{1}}{2}, \frac{y_{2} + y_{1}}{2}\right) \) - \( M = \left(x_{1} + \frac{x_{2} - x_{1}}{2}, y_{1} + \frac{y_{2} - y_{1}}{2}\right) \) **Both the second and third formulas are correct.**