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The problem is asking to find the value of \( x \) in the given geometric figure, where \( O \) is the center of the circle, and points \( A \), \( B \), and \( C \) are on the circle. Additionally, the angle at point \( A \) is given as \( 50^\circ \).

This type of problem involves the use of **circle theorems**. In particular:

- **Angle at the center theorem**: The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circle's circumference.
  
Here, angle \( \angle AOB \) is subtended at the center, and angle \( \angle ACB \) is subtended at the circumference.

### Step-by-step solution:
1. We are given that \( \angle A = 50^\circ \). This is the angle at point \( A \), subtended by the arc \( BC \) on the circumference.
   
2. According to the angle at the center theorem:
   \[
   \angle BOC = 2 \times \angle BAC
   \]
   So:
   \[
   \angle BOC = 2 \times 50^\circ = 100^\circ
   \]

3. In the triangle \( \triangle BOC \), the angle at point \( O \), denoted as \( x \), is the angle subtended by arc \( BC \) at the center. Since the angle subtended by the same arc at the circumference \( A \) is \( 50^\circ \), the angle subtended at the center is \( 100^\circ \).

Thus, the value of \( x = 100^\circ \).

Would you like any further details or have any questions? 

### Related questions:
1. What other theorems are commonly used in circle geometry problems?
2. How can we prove the angle at the center theorem?
3. How do we calculate the circumference of a circle given its radius?
4. How is the sum of interior angles in a triangle derived?
5. What is the relationship between inscribed and central angles in a semicircle?

### Tip:
In problems involving circles, always consider the various theorems related to angles, chords, and tangents, as they can simplify many seemingly complex problems.

Find the value of x in the following figure where O is the centre of the given circle and A, B & C are points on the circle. The angle at A is 50 degrees.

Learn how to find the value of x in a circle geometry problem using the Angle at the Center Theorem. Given a 50-degree angle at the circumference, apply the circle theorem to calculate the angle subtended at the center. This solution is suitable for Grades 8-10.

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