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Let's analyze the problem presented in the image and solve it.

### Problem Recap:
- You are given two points \( A(-2, 3) \) and \( B(4, -2) \).
- The question asks for the equation of the straight line passing through the point \( P(1, 1) \) that is perpendicular to the line \( AB \).

### Solution:

1. **Find the Slope of Line \( AB \)**:
   - The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
   \[
   m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   - For points \( A(-2, 3) \) and \( B(4, -2) \), the slope \( m_{AB} \) is:
   \[
   m_{AB} = \frac{-2 - 3}{4 - (-2)} = \frac{-5}{6}
   \]

2. **Find the Slope of the Perpendicular Line**:
   - The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
   - If \( m \) is the slope of a line, the slope of a line perpendicular to it is \( -\frac{1}{m} \).
   - Therefore, the slope of the line perpendicular to \( AB \) is:
   \[
   m_{\perp} = -\frac{1}{-\frac{5}{6}} = \frac{6}{5}
   \]

3. **Write the Equation of the Line through \( P(1, 1) \) with Slope \( \frac{6}{5} \)**:
   - The equation of a line with slope \( m \) passing through a point \( (x_1, y_1) \) is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   - Substituting \( m = \frac{6}{5} \), \( x_1 = 1 \), and \( y_1 = 1 \), we get:
   \[
   y - 1 = \frac{6}{5}(x - 1)
   \]
   - Simplifying the equation:
   \[
   y - 1 = \frac{6}{5}x - \frac{6}{5}
   \]
   \[
   y = \frac{6}{5}x - \frac{6}{5} + 1
   \]
   \[
   y = \frac{6}{5}x - \frac{1}{5}
   \]

Thus, the equation of the line passing through \( P(1, 1) \) and perpendicular to \( AB \) is:
\[
y = \frac{6}{5}x - \frac{1}{5}
\]

Would you like further explanation or have any other questions?

### Related Questions:
1. How do you find the equation of a line parallel to a given line passing through a specific point?
2. Can you determine the point of intersection between the line you just found and the line \( AB \)?
3. What is the distance between point \( P(1, 1) \) and the line \( AB \)?
4. How would the equation change if the line was parallel to \( AB \) instead of perpendicular?
5. How do you find the equation of a line in general form given two points?

### Tip:
Always double-check the signs when calculating slopes and remember that the perpendicular slope is the negative reciprocal. This is a common area where errors can occur.

Learn how to find the equation of a line perpendicular to AB passing through the point P(1, 1) with a step-by-step solution. This problem involves calculating slopes, using the negative reciprocal rule, and applying the point-slope form of a linear equation. Suitable for high school students in Grades 9-12.

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