The problem shows a circle with center $O$, and a diagram involving a straight line $AN$, with points $A$, $B$, and $C$ along the line and the circle. There are two key angles shown: one labeled as $30^\circ$, and variables $x$ and $h$ that we need to solve for.

Here's the step-by-step process to find the values of $x$ and $h$, based on what we can infer from the image:

1. Analyze the Given Information

• $O$ is the center of the circle.
• $AN$ is a straight line.
• The angle $\angle AOB$ is $30^\circ$.
• $OB$ and $OC$ are radii of the circle, implying $OB = OC = r$, where $r$ is the radius of the circle.
• $AB$ is given as a distance on the line, and we need to find the values for $x$ (which may be the distance from $A$ to $B$) and $h$, which seems to be the height or vertical distance from $B$ to $C$.

2. Using Geometry of Circles

• Since $O$ is the center, and $OB$ and $OC$ are radii, triangle $OBC$ may be an isosceles triangle.
• The angle at $O$ ($\angle AOB = 30^\circ$) provides useful information for calculating the chord length or position of point $B$.
• If $AN$ is a tangent, we can use properties of tangents and radii for further calculations.

3. Applying Trigonometry

• If we assume $AB = x$, and $\angle AOB = 30^\circ$, then we can calculate the chord length $AB$ using: $AB = 2r \sin\left(\frac{\angle AOB}{2}\right) = 2r \sin(15^\circ)$ Hence, $x = AB$ could be calculated from this.

• For $h$, which might represent the perpendicular height from $B$ to the line, we can use basic trigonometric relationships in the right triangle formed by the radius and line segment.

Further Steps

• To solve this exactly, we need either the radius $r$ or another piece of data related to the distances.

Would you like me to assume a value for the radius and proceed with an example solution? Or do you have additional data for $r$ or another distance? Let me know how you'd like to proceed!

Expanding Your Understanding

1. What is the significance of $\angle AOB$ being $30^\circ$?
2. How does the property of radii being equal influence triangle $OBC$?
3. What is the geometric relationship between a chord and a radius of a circle?
4. How can trigonometry help in solving problems involving angles and circles?
5. Can you identify which triangle in the diagram is a right triangle?

Tip: Always consider properties of tangent lines when working with circles, as they are perpendicular to the radius at the point of contact.