Solution:

Given:
Point A = (4,3)
Point B = (x,5)
Center O = (2,3)

To find the value of x, we will use the distance formula and the properties of a circle.

Distance formula:
The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Using the distance formula, we can find the distance between points A and O and between points B and O.

1. Finding the distance between point A and O:
d₁ = √((2 - 4)² + (3 - 3)²)
= √((-2)² + 0²)
= √(4 + 0)
= √4
= 2

2. Finding the distance between point B and O:
d₂ = √((2 - x)² + (3 - 5)²)
= √((2 - x)² + (-2)²)
= √((2 - x)² + 4)
= √(4 - 4x + x² + 4)
= √(x² - 4x + 8)

Properties of a circle:
For any point on a circle, the distance between that point and the center of the circle is equal to the radius of the circle.

3. Equating the distances:
Since both point A and point B are on the circle with center O, their distances from O should be equal to the radius of the circle.

d₁ = d₂
2 = √(x² - 4x + 8)

Squaring both sides of the equation to remove the square root:

4 = x² - 4x + 8

Rearranging the equation:

x² - 4x + 4 = 0

Factoring the quadratic equation:

(x - 2)(x - 2) = 0

(x - 2)² = 0

Taking the square root of both sides:

x - 2 = 0

x = 2

Therefore, the value of x is 2.

Conclusion:
The value of x is 2.