The radius of the circle passing through the points `(1, 2), (5,2) and (5,-2)` is

Options

A. `2sqrt(2)`
B. `3sqrt(2)`
C. `2sqrt(2)`
D. `5sqrt(2)`

Answer

To find the radius of the circle passing through the points (1, 2), (5, 2), and (5, -2), we can follow these steps:

### Step 1: Identify the points
Let the points be:
- A(1, 2)
- B(5, 2)
- C(5, -2)

### Step 2: Find the center of the circle
We know that the distances from the center of the circle to each of these points must be equal. Let's denote the center of the circle as O(x, y).

### Step 3: Set up the equations
Using the distance formula, we can set up the equations based on the distances from the center O to points A and B:
1. Distance OA: \( OA^2 = (x - 1)^2 + (y - 2)^2 \)
2. Distance OB: \( OB^2 = (x - 5)^2 + (y - 2)^2 \)

Since OA = OB, we can equate these two distances:
\[ (x - 1)^2 + (y - 2)^2 = (x - 5)^2 + (y - 2)^2 \]

### Step 4: Simplify the equation
By simplifying the equation, we can eliminate \( (y - 2)^2 \):
\[ (x - 1)^2 = (x - 5)^2 \]
Expanding both sides:
\[ x^2 - 2x + 1 = x^2 - 10x + 25 \]
Now, cancel \( x^2 \) from both sides:
\[ -2x + 1 = -10x + 25 \]
Rearranging gives:
\[ 8x = 24 \]
Thus,
\[ x = 3 \]

### Step 5: Find the y-coordinate
Now we need to find the y-coordinate. We can use the distance from the center O to point B(5, 2):
\[ OB^2 = (3 - 5)^2 + (y - 2)^2 \]
We also know that the distance from O to point C(5, -2) must be equal to OB:
\[ OC^2 = (3 - 5)^2 + (y + 2)^2 \]

Setting these equal:
\[ (3 - 5)^2 + (y - 2)^2 = (3 - 5)^2 + (y + 2)^2 \]
Simplifying gives:
\[ (y - 2)^2 = (y + 2)^2 \]
Expanding both sides:
\[ y^2 - 4y + 4 = y^2 + 4y + 4 \]
Canceling \( y^2 + 4 \) from both sides:
\[ -4y = 4y \]
This leads to:
\[ 8y = 0 \implies y = 0 \]

### Step 6: Determine the center of the circle
Thus, the center of the circle is at \( O(3, 0) \).

### Step 7: Calculate the radius
Now we can calculate the radius using the distance from the center O to point B(5, 2):
\[ r = \sqrt{(5 - 3)^2 + (2 - 0)^2} \]
Calculating this gives:
\[ r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Conclusion
The radius of the circle passing through the points (1, 2), (5, 2), and (5, -2) is \( 2\sqrt{2} \).

---