The problem in the image involves a right triangle inscribed in a circle, with angle $C = 67^\circ$, distance $x = 27$, and the task is to find the radius $r$ of the circle, rounding the answer to the nearest whole number.

Approach:

1. Right Triangle Geometry: The points $C$, $B$, and $D$ form a right triangle since the radius $CD$ is perpendicular to the tangent line at point $D$. The right angle at $D$ helps use trigonometric relationships.

2. Trigonometric Relationships: We can apply trigonometry using the given angle at $C$, $x = 27$ as the hypotenuse (distance from $C$ to $A$), and $r$ as one of the sides.

   • Use the cosine of the angle $C$ to relate $r$ and $x$.

$\cos(C) = \frac{r}{x}$

$r = x \cdot \cos(C)$

3. Substitute values:
   • $C = 67^\circ$
   • $x = 27$

Now I will calculate $r$.The radius $r$ of the circle is approximately $11$ when rounded to the nearest whole number.

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

1. How does the sine or tangent of an angle relate to the sides of a triangle?
2. How would you find the radius if a different angle at $C$ was given?
3. What other trigonometric functions could be used in this problem?
4. How does the Pythagorean theorem apply in a right triangle within a circle?
5. What other geometric properties can be derived from a circle with an inscribed triangle?

Tip: Always double-check if a triangle is right-angled before using basic trigonometric identities, as this ensures correct application.