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Let's solve question 13.

Given points:
- \( A(-7, 7) \)
- \( B(1, 9) \)
- \( C(3, 1) \)
- \( D(-7, 1) \)

These points lie on a circle, and lines \( AB \) and \( CD \) are chords of the circle.

### Question 13a: Find the equation of the perpendicular bisector of \( AB \).

1. **Midpoint of \( AB \)**:
   - The midpoint \( M \) of \( AB \) can be found using the midpoint formula:
     \[
     M = \left( \frac{-7 + 1}{2}, \frac{7 + 9}{2} \right) = \left( -3, 8 \right)
     \]

2. **Slope of \( AB \)**:
   - The slope \( m_{AB} \) of line \( AB \) is:
     \[
     m_{AB} = \frac{9 - 7}{1 - (-7)} = \frac{2}{8} = \frac{1}{4}
     \]

3. **Slope of the perpendicular bisector**:
   - The perpendicular bisector will have a slope that is the negative reciprocal of \( \frac{1}{4} \), which is \( -4 \).

4. **Equation of the perpendicular bisector**:
   - Using the point-slope form \( y - y_1 = m(x - x_1) \) at midpoint \( M(-3, 8) \):
     \[
     y - 8 = -4(x + 3)
     \]
   - Simplifying:
     \[
     y = -4x - 12 + 8
     \]
     \[
     y = -4x - 4
     \]
   - Therefore, the equation of the perpendicular bisector of \( AB \) is:
     \[
     y = -4x - 4
     \]

### Question 13b: Find the coordinates of the center of the circle.

Since \( AB \) and \( CD \) are chords of the circle, the center of the circle lies on the perpendicular bisectors of both \( AB \) and \( CD \). To find the center, we’ll also need the perpendicular bisector of \( CD \).

1. **Midpoint of \( CD \)**:
   - The midpoint \( N \) of \( CD \) is:
     \[
     N = \left( \frac{-7 + 3}{2}, \frac{1 + 1}{2} \right) = \left( -2, 1 \right)
     \]

2. **Slope of \( CD \)**:
   - The slope \( m_{CD} \) of \( CD \) is:
     \[
     m_{CD} = \frac{1 - 1}{3 - (-7)} = \frac{0}{10} = 0
     \]

3. **Slope of the perpendicular bisector of \( CD \)**:
   - Since the slope of \( CD \) is \( 0 \), it’s a horizontal line, so its perpendicular bisector is a vertical line.
   - The equation of the perpendicular bisector of \( CD \) is:
     \[
     x = -2
     \]

4. **Finding the intersection of the perpendicular bisectors**:
   - We have two equations:
     \[
     y = -4x - 4
     \]
     \[
     x = -2
     \]
   - Substitute \( x = -2 \) into \( y = -4x - 4 \):
     \[
     y = -4(-2) - 4 = 8 - 4 = 4
     \]
   - Therefore, the center of the circle is at:
     \[
     (-2, 4)
     \]

### Final Answers:

- **13a**: The equation of the perpendicular bisector of \( AB \) is \( y = -4x - 4 \).
- **13b**: The coordinates of the center of the circle are \( (-2, 4) \).

Would you like further details, or do you have any questions?

### Related Questions:
1. How would you find the radius of the circle if the center is known?
2. What is the general form of a circle's equation when the center and radius are known?
3. How can you verify that all four points lie on the same circle?
4. What changes if the slopes of \( AB \) and \( CD \) were equal?
5. How do perpendicular bisectors help in finding the center of a circle formed by multiple points?

#### Tip:
Remember, the perpendicular bisector of a line segment always passes through the midpoint of the segment and has a slope that is the negative reciprocal of the segment's slope.

Step-by-step solution for Problem 13, finding the perpendicular bisectors of chords AB and CD, and determining the center of the circle formed by points A(-7, 7), B(1, 9), C(3, 1), and D(-7, 1). Suitable for Grades 10-12.

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