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Let's break down each part of the problem presented in the image:

### a. Refer back to the graph of \( x^2 + y^2 = 25 \). What is the radius? How is the radius of the circle related to the equation?

This is the equation of a circle in the form \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle.
- Here, \( r^2 = 25 \), so \( r = \sqrt{25} = 5 \).
- **Answer**: The radius of the circle is 5. In the equation \( x^2 + y^2 = r^2 \), the radius is the square root of the constant on the right-hand side.

### b. What would be the equation of a circle that has its center at \( (5, -7) \) with radius 10? With radius 12?

The general equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- For center \( (5, -7) \) and radius 10:  
\[
(x - 5)^2 + (y + 7)^2 = 10^2 = 100
\]
The equation is \( (x - 5)^2 + (y + 7)^2 = 100 \).

- For center \( (5, -7) \) and radius 12:  
\[
(x - 5)^2 + (y + 7)^2 = 12^2 = 144
\]
The equation is \( (x - 5)^2 + (y + 7)^2 = 144 \).

### c. Now generalize the connection between the radius and the equation of a circle. Write a general equation for a circle with any center \( (h, k) \) and radius \( r \).

The general equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Here, \( (h, k) \) represents the coordinates of the center, and \( r \) is the radius. The radius is obtained as the square root of the constant on the right-hand side of the equation.

### d. Given the equation \( (x - 3)^2 + (y + 7)^2 = 169 \), how can you find the radius of the circle?

This equation is already in the standard form of a circle equation:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Here, \( h = 3 \), \( k = -7 \), and \( r^2 = 169 \).
- To find the radius, take the square root of 169:  
\[
r = \sqrt{169} = 13
\]
- **Answer**: The radius of the circle is 13.

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Do you want further details or explanations? Here are some related questions:

1. How can you derive the equation of a circle if given the center and a point on the circumference?
2. If you know the diameter of a circle, how can you use it to write the circle's equation?
3. How would the equation of a circle change if the center is at the origin \( (0, 0) \)?
4. What is the geometric significance of each term in the equation \( (x - h)^2 + (y - k)^2 = r^2 \)?
5. How do you convert the general form of a circle \( Ax^2 + Ay^2 + Dx + Ey + F = 0 \) into the standard form?

**Tip**: When writing the equation of a circle, always remember that \( (h, k) \) represents the center, and \( r \) is the radius, which can be calculated by taking the square root of the right-hand side.

A circle has a special characteristic, its radius, which defines its size.

 a. Refer back to the graph of x^2 + y^2 = 25. What is the radius? How is the radius of the circle related to the equation?
 b. What would be the equation of a circle that has its center at (5, -7) with radius 10? With radius 12?
 c. Now generalize the connection between the radius and the equation of a circle. Write a general equation for a circle with any center (h, k) and radius r.
 d. Given the equation (x - 3)^2 + (y + 7)^2 = 169, how can you find the radius of the circle?

Explore how to find the radius of a circle from its equation, including examples of circles centered at different points. This guide covers circle equations, including x^2 + y^2 = r^2 and (x - h)^2 + (y - k)^2 = r^2, and is perfect for students in grades 8-10.

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