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The image contains a geometry problem regarding triangle \( \Delta ABC \), where the coordinates of the vertices are given as:

- \( A(-14, 6) \)
- \( B(3, 13) \)
- \( C(-7, -11) \)

The tasks are:

1. Find the three perpendicular bisectors of the sides of the triangle.
2. Find the circumcenter.
3. Graph and draw the circumcircle, and then find its radius.

### Step-by-step solution:

1. **Finding the perpendicular bisectors:**
   To find the perpendicular bisectors, you need the midpoint of each side and the slope of each side. The perpendicular bisector will have a slope that is the negative reciprocal of the slope of the side and pass through the midpoint.

   - **Side \( AB \):**
     - Midpoint: \( \left( \frac{-14 + 3}{2}, \frac{6 + 13}{2} \right) = \left( \frac{-11}{2}, \frac{19}{2} \right) = (-5.5, 9.5) \)
     - Slope of \( AB \): \( m = \frac{13 - 6}{3 - (-14)} = \frac{7}{17} \)
     - Perpendicular slope: \( -\frac{17}{7} \)
     - Equation: Using the point-slope form: \( y - 9.5 = -\frac{17}{7}(x + 5.5) \)

   - **Side \( BC \):**
     - Midpoint: \( \left( \frac{3 + (-7)}{2}, \frac{13 + (-11)}{2} \right) = (-2, 1) \)
     - Slope of \( BC \): \( m = \frac{-11 - 13}{-7 - 3} = \frac{-24}{-10} = 2.4 \)
     - Perpendicular slope: \( -\frac{1}{2.4} = -\frac{5}{12} \)
     - Equation: Using the point-slope form: \( y - 1 = -\frac{5}{12}(x + 2) \)

   - **Side \( AC \):**
     - Midpoint: \( \left( \frac{-14 + (-7)}{2}, \frac{6 + (-11)}{2} \right) = (-10.5, -2.5) \)
     - Slope of \( AC \): \( m = \frac{-11 - 6}{-7 - (-14)} = \frac{-17}{7} \)
     - Perpendicular slope: \( \frac{7}{17} \)
     - Equation: Using the point-slope form: \( y + 2.5 = \frac{7}{17}(x + 10.5) \)

2. **Finding the circumcenter:**
   The circumcenter is the point where the perpendicular bisectors of the triangle meet. Solving the system of equations from the bisectors of sides \( AB \) and \( BC \) will give the circumcenter. This requires solving for the intersection of these lines.

3. **Graphing the circumcircle and finding the radius:**
   Once the circumcenter is known, the circumradius is the distance from the circumcenter to any vertex of the triangle. This can be calculated using the distance formula.

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Here are some relative questions:

1. How do you find the midpoint of a line segment in a coordinate plane?
2. What is the significance of the circumcenter in a triangle?
3. How is the equation of a line derived using point-slope form?
4. What does it mean for two lines to be perpendicular in terms of their slopes?
5. How can you verify if the circumcenter is correct by graphing?

**Tip:** 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.

Homework: Using ΔABC: A(-14,6), B(3,13), C(-7,-11).
1. Find the 3 perpendicular bisectors.
2. Find the circumcenter.
3. Graph and draw the circumcircle. Then find its radius.

Learn how to calculate the perpendicular bisectors, circumcenter, and circumcircle for the triangle with vertices A(-14,6), B(3,13), and C(-7,-11). This problem explores key geometry concepts such as the midpoint formula, slope, and distance calculation. Suitable for students in Grades 9-12.

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