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The image contains a geometry problem in Persian with two points, \( A(7, 2) \) and \( B(6, -5) \), and has several parts. Here's a breakdown of the problem:

1. **Part (الف)**: Find the distances of \( OA \), \( OB \), and \( AB \).  
   (Where \( O \) likely refers to the origin \( (0, 0) \)).

2. **Part (ب)**: What are the coordinates of the midpoint of \( A \) and \( B \)?

3. **Part (پ)**: What is the equation of the perpendicular bisector of segment \( AB \)?

### Let's solve each part.

#### Part 1: Distances of \( OA \), \( OB \), and \( AB \)
- Distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
  \[
  d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  \]
  
- **Distance \( OA \)** (from the origin to \( A(7, 2) \)):
  \[
  OA = \sqrt{(7 - 0)^2 + (2 - 0)^2} = \sqrt{49 + 4} = \sqrt{53}
  \]

- **Distance \( OB \)** (from the origin to \( B(6, -5) \)):
  \[
  OB = \sqrt{(6 - 0)^2 + (-5 - 0)^2} = \sqrt{36 + 25} = \sqrt{61}
  \]

- **Distance \( AB \)** (between \( A(7, 2) \) and \( B(6, -5) \)):
  \[
  AB = \sqrt{(6 - 7)^2 + (-5 - 2)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}
  \]

#### Part 2: Midpoint of \( A \) and \( B \)
- The midpoint formula is:
  \[
  M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  \]
  For \( A(7, 2) \) and \( B(6, -5) \):
  \[
  M = \left( \frac{7 + 6}{2}, \frac{2 + (-5)}{2} \right) = \left( \frac{13}{2}, \frac{-3}{2} \right)
  \]

#### Part 3: Equation of the Perpendicular Bisector of \( AB \)
1. First, the slope of \( AB \) is:
   \[
   m_{AB} = \frac{-5 - 2}{6 - 7} = \frac{-7}{-1} = 7
   \]
2. The perpendicular slope is the negative reciprocal:
   \[
   m_{\perp} = -\frac{1}{7}
   \]
3. The perpendicular bisector passes through the midpoint \( M \left( \frac{13}{2}, \frac{-3}{2} \right) \), so the equation is:
   \[
   y - \left( \frac{-3}{2} \right) = -\frac{1}{7} \left( x - \frac{13}{2} \right)
   \]
   Simplifying this gives the equation of the perpendicular bisector.

Would you like me to explain any step in more detail?

Here are five related questions:
1. How do you calculate the distance between two points in 3D?
2. What is the difference between the midpoint and centroid of a triangle?
3. How can you find the equation of a line given two points?
4. What are the properties of the perpendicular bisector in a triangle?
5. How does the slope of a line relate to its angle with the x-axis?

**Tip:** Always keep track of signs when working with coordinates to avoid errors in distance and midpoint calculations.

Find distances of OA, OB, AB, midpoint between A and B, and the equation of the perpendicular bisector of AB for points A(7,2) and B(6,-5).

Solve a coordinate geometry problem involving distances OA, OB, AB, the midpoint, and the equation of the perpendicular bisector of AB for points A(7, 2) and B(6, -5). Includes key concepts like distance formula, midpoint formula, and perpendicular bisector.

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