Given information:
- Point A is located at (4, 3).
- Point B is located at (x, 5).
- The center of the circle is O, which is located at (2, 3).

Approach:
To find the value of x, we can use the distance formula to calculate the distance between points A and O, as well as points B and O. Since the distance between a point and the center of a circle is equal to the radius of the circle, we can equate these distances and solve for x.

Calculation:
Step 1: Find the distance between points A and O.
Using the distance formula, we have:
Distance AO = √[(x2 - x1)² + (y2 - y1)²]
= √[(2 - 4)² + (3 - 3)²]
= √[(-2)² + 0²]
= √[4 + 0]
= √4
= 2

Step 2: Find the distance between points B and O.
Using the distance formula, we have:
Distance BO = √[(x2 - x1)² + (y2 - y1)²]
= √[(x - 2)² + (5 - 3)²]
= √[(x - 2)² + 2²]
= √[x² - 4x + 4 + 4]
= √[x² - 4x + 8]

Step 3: Equate the distances AO and BO.
Since the distances AO and BO are equal (as they represent the radius of the circle), we can set up the following equation:
2 = √[x² - 4x + 8]

Step 4: Solve the equation.
To solve the equation, we need to isolate x² - 4x + 8 on one side of the equation. Squaring both sides of the equation, we get:
4 = x² - 4x + 8

Now rearrange the equation:
x² - 4x + 4 = 0

This equation can be factored as:
(x - 2)(x - 2) = 0

Therefore, x - 2 = 0
x = 2

Answer:
The value of x is 2.