To determine the value of $x$ in the given diagram, we can use the properties of angles subtended by a chord and the properties of tangents to a circle.

Given:

• The angle outside the circle is $58^\circ$.
• $x$ is the angle subtended at the center of the circle by the chord.

Step-by-step solution:

1. Identify the angle at the circumference: The angle outside the circle, $58^\circ$, is between the tangent to the circle and the chord.

2. Tangent-Chord Angle Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Therefore, the angle in the alternate segment, which is the angle subtended by the chord at the circumference of the circle (not at the center), is also $58^\circ$.

3. Relationship between angle at the center and angle at the circumference: The angle subtended by a chord at the center of the circle is twice the angle subtended at the circumference by the same chord.

Therefore: $x = 2 \times 58^\circ$

So, we have: $x = 116^\circ$

Thus, the value of $x$ is $116^\circ$.

Would you like any more details or have any questions? Here are five questions you might consider asking:

1. How does the Tangent-Chord Angle Theorem help in solving problems involving circles?
2. What is the relationship between angles subtended at the center and at the circumference?
3. Can you explain other circle theorems useful in geometry?
4. How can we determine the measure of angles formed by tangents and chords in a circle?
5. What is the significance of inscribed angles in a circle?

Tip: Remember, the angle subtended by a chord at the center of the circle is always twice the angle subtended by the same chord at the circumference. This is a crucial concept in circle geometry.