The coordinates of the points $$ A $$ and $$ B $$ are $$ (-15,10) $$ and $$ (5,-6) $$ respectively. Find the equation of the perpendicular bisector of $$ A B $$. Give your answer in the form $$ a x+b y=c $$ where $$ a, b $$ and $$ c $$ are integers in their lowest terms. .

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To find the equation of the perpendicular bisector of segment $$ AB $$ where $$ A(-15, 10) $$ and $$ B(5, -6) $$, follow these steps:

1. **Find the Midpoint $$ M $$ of $$ AB $$:**
   $$
   M = \left( \frac{-15 + 5}{2}, \frac{10 + (-6)}{2} \right) = \left( \frac{-10}{2}, \frac{4}{2} \right) = (-5, 2)
   $$

2. **Calculate the Slope of $$ AB $$:**
   $$
   m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 10}{5 - (-15)} = \frac{-16}{20} = -\frac{4}{5}
   $$

3. **Determine the Slope of the Perpendicular Bisector:**
   The perpendicular slope is the negative reciprocal of $$ m_{AB} $$:
   $$
   m_{\perp} = \frac{5}{4}
   $$

4. **Use the Point-Slope Form with Midpoint $$ M $$ and Slope $$ m_{\perp} $$:**
   $$
   y - 2 = \frac{5}{4}(x + 5)
   $$
   
5. **Convert to the Standard Form $$ ax + by = c $$:**
   Multiply both sides by 4 to eliminate the fraction:
   $$
   4(y - 2) = 5(x + 5) \
   4y - 8 = 5x + 25 \
   -5x + 4y = 33
   $$
   
   To ensure the coefficient of $$ x $$ is positive, multiply the entire equation by -1:
   $$
   5x - 4y = -33
   $$

**Final Answer:**
$$
5\,x\; -\;4\,y\;=\;-33
$$
The equation of the perpendicular bisector of $$ AB $$ is $$ 5x - 4y = -33 $$.

The coordinates of the points and are and
respectively.
Find the equation of the perpendicular bisector of .
Give your answer in the form
where and are integers in their lowest terms.
.

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The coordinates of the points and are and respectively. Find the equation of the perpendicular bisector of . Give your answer in the form where and are integers in their lowest terms. .