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. $$ M=\left(\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}\right) $$ $$ M=\left(\frac{x_{2}+x_{1}}{2}, \frac{y_{2}+y_{1}}{2}\right) $$ $$ M=\left(x_{1}+\frac{x_{2}-x_{1}}{2}, y_{1}+\frac{y_{2}-y_{1}}{2}\right) $$ $$ M=\left(x_{2}-x_{1}, y_{2}-y_{1}\right) $$

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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.**
The correct formulas to find the midpoint are: - $$ M = \left(\frac{x_{2} + x_{1}}{2}, \frac{y_{2} + y_{1}}{2}\right) $$ - $$ M = \left(x_{1} + \frac{x_{2} - x_{1}}{2}, y_{1} + \frac{y_{2} - y_{1}}{2}\right) $$

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