Consider the following equation of a circle.

x2+y2−6x−8y+16=0
Step 2 of 3: Find the radius, r.

To find the radius of the circle given the equation \( x^2 + y^2 - 6x - 8y + 16 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

To convert the given equation to this form, we'll complete the square for both \(x\) and \(y\).

### Step 2: Complete the square
Given equation:

\[
x^2 + y^2 - 6x - 8y + 16 = 0
\]

First, group the \(x\) and \(y\) terms:

\[
(x^2 - 6x) + (y^2 - 8y) + 16 = 0
\]

Next, complete the square for \(x\) and \(y\):

1. For \(x^2 - 6x\), add and subtract \(\left(\frac{6}{2}\right)^2 = 9\):
   \[
   (x^2 - 6x + 9) - 9
   \]
   This becomes \((x - 3)^2 - 9\).

2. For \(y^2 - 8y\), add and subtract \(\left(\frac{8}{2}\right)^2 = 16\):
   \[
   (y^2 - 8y + 16) - 16
   \]
   This becomes \((y - 4)^2 - 16\).

Now substitute back:

\[
(x - 3)^2 - 9 + (y - 4)^2 - 16 + 16 = 0
\]

Simplify:

\[
(x - 3)^2 + (y - 4)^2 - 9 = 0
\]

Add 9 to both sides:

\[
(x - 3)^2 + (y - 4)^2 = 9
\]

### Step 3: Identify the radius
From the equation \((x - 3)^2 + (y - 4)^2 = 9\), we see that the radius squared \(r^2 = 9\), so the radius is:

\[
r = \sqrt{9} = 3
\]

**The radius \(r\) is 3.**

Would you like more details on any of these steps or have any other questions?

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Here are 5 related questions:
1. How do you find the center of the circle given the same equation?
2. Can you explain the process of completing the square in more detail?
3. How would the equation change if the circle was shifted horizontally by 2 units?
4. What is the significance of the standard form of a circle?
5. How can you determine the diameter of the circle from the given equation?

**Tip:** When completing the square, always remember to balance the equation by adding or subtracting the same value on both sides.

Discover how to find the radius of a circle with the equation x^2 + y^2 - 6x - 8y + 16 = 0. This detailed step-by-step guide covers completing the square and identifying key geometric properties. Suitable for high school students studying circle equations and geometry.

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