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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.

If C = 67° and x = 27, find r. (Round your answer to the nearest whole number.)

Learn how to find the radius of a circle using trigonometry and right triangle geometry. This problem involves applying the cosine function to determine the radius based on a given angle and distance. Suitable for students in Grades 9-11.

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