The problem asks us to find the value of $x$ in the given circle. We are given that the angle at $T$ is $52^{\circ}$. The lines $RS$ and $TS$ are radii of the circle, so they have the same length.

1. Problem Description

The problem asks us to find the value of $x$ in the given circle. We are given that the angle at $T$ is $52^{\circ}$. The lines $RS$ and $TS$ are radii of the circle, so they have the same length.

2. Solution Steps

Since $RS$ and $TS$ are radii of the circle, triangle $RST$ is an isosceles triangle.

Therefore, the angles at $R$ and $T$ are equal, so $\angle SRT = \angle RTS = 52^{\circ}$.

The sum of angles in a triangle is $180^{\circ}$, so $\angle RST = 180^{\circ} - \angle SRT - \angle RTS = 180^{\circ} - 52^{\circ} - 52^{\circ} = 180^{\circ} - 104^{\circ} = 76^{\circ}$.

The measure of an inscribed angle is half the measure of the intercepted arc.

So, the measure of arc $RT$ is $2 \times \angle RST = 2 \times 76^{\circ} = 152^{\circ}$.

The angle $x$ is an inscribed angle that intercepts the arc $RT$.

Thus, $x = \frac{1}{2} \times \text{arc } RT$. The inscribed angle that intercepts the arc RT is $\angle RST$ which is 76 degrees. It cannot be $x$.

Instead, the question asks for the measure of arc ST. $x$ is the arc $RS$. The arc ST has measure $2 * \angle SRT = 2 * 52 = 104$

We can use the circumference of the circle $360 = x + 104 + 152$

$360 = x + 256$

$x = 360 - 256 = 104$

3. Final Answer

The value of x is 104.