7. Determine if $$ M $$ lies on the perpendicular bisector of $$ \overline{A B} $$, given: $$ A(-3,2), B(5,6) $$ and $$ M(1,2) $$.

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To determine whether point $$ M(1, 2) $$ lies on the **perpendicular bisector** of the line segment $$ \overline{AB} $$ with endpoints $$ A(-3, 2) $$ and $$ B(5, 6) $$, follow these steps:

### 1. Find the Midpoint of $$ \overline{AB} $$

The midpoint $$ M_{\text{mid}} $$ of $$ \overline{AB} $$ is calculated as:

$$
M_{\text{mid}} = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{-3 + 5}{2}, \frac{2 + 6}{2} \right) = (1, 4)
$$

### 2. Determine the Slope of $$ \overline{AB} $$

$$
\text{Slope of } AB = \frac{y_B - y_A}{x_B - x_A} = \frac{6 - 2}{5 - (-3)} = \frac{4}{8} = 0.5
$$

### 3. Find the Slope of the Perpendicular Bisector

The perpendicular bisector has a slope that is the **negative reciprocal** of the slope of $$ \overline{AB} $$:

$$
\text{Slope of perpendicular bisector} = -\frac{1}{0.5} = -2
$$

### 4. Write the Equation of the Perpendicular Bisector

Using the point-slope form with midpoint $$ (1, 4) $$ and slope $$ -2 $$:

$$
y - 4 = -2(x - 1) \
y = -2x + 2 + 4 \
y = -2x + 6
$$

### 5. Check if $$ M(1, 2) $$ Lies on the Perpendicular Bisector

Plug $$ x = 1 $$ into the equation:

$$
y = -2(1) + 6 = 4
$$

However, the $$ y $$-coordinate of point $$ M $$ is **2**, not **4**.

### 6. Alternative Verification: Check Distances

Another way to confirm is to check if $$ M $$ is equidistant from $$ A $$ and $$ B $$:

$$
\text{Distance } MA = \sqrt{(1 - (-3))^2 + (2 - 2)^2} = \sqrt{4^2 + 0^2} = 4
$$

$$
\text{Distance } MB = \sqrt{(1 - 5)^2 + (2 - 6)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66
$$

Since $$ MA \neq MB $$, point $$ M $$ is **not** equidistant from $$ A $$ and $$ B $$, confirming it does **not** lie on the perpendicular bisector.

### **Conclusion**

Point $$ M(1, 2) $$ **does not** lie on the perpendicular bisector of $$ \overline{AB} $$.

**Answer:**  
No, point M (1, 2) does not lie on the perpendicular bisector of segment AB.
Point $$ M(1, 2) $$ does not lie on the perpendicular bisector of $$ \overline{AB} $$.

7. Determine if \( M \) lies on the perpendicular bisector of \( \overline{A B} \), given: \( A(-3,2), B(5,6) \) and \( M(1,2) \).

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