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The given diagram shows a circle with center \( O \) and radius \( r \). Point \( A \) is a point on the circle, and the segment \( AB \) is a tangent to the circle at point \( A \). The angle \( \angle OAB = 37^\circ \), and the length of \( AB \) is given as 8 units. We need to find the radius \( r \) of the circle.

### Key Information:
- \( AB = 8 \) units.
- \( \angle OAB = 37^\circ \).
- \( OA = r \) (radius of the circle).
- \( OB \) is perpendicular to \( AB \), making \( \triangle OAB \) a right triangle.

We can use trigonometric relations in the right triangle \( OAB \). Specifically, the tangent of the angle \( 37^\circ \) can be used because:
\[
\tan(37^\circ) = \frac{AB}{OA} = \frac{8}{r}.
\]
Thus, we can solve for \( r \):
\[
r = \frac{8}{\tan(37^\circ)}.
\]

Let me calculate that.The radius \( r \) of the circle is approximately 10.62 units.

Would you like further details or have any other questions?

Here are 5 related questions you might find interesting:
1. How does the tangent function relate to other angles in right triangles?
2. What would happen if the angle \( \angle OAB \) was increased to \( 45^\circ \)?
3. How would the radius change if \( AB \) were doubled while keeping the same angle?
4. Can you derive the formula for \( r \) using another trigonometric function, such as sine or cosine?
5. How do tangents of circles differ in geometry problems involving secants?

**Tip:** Always remember that the tangent function in a right triangle is the ratio of the opposite side to the adjacent side.

Calculate the radius 'r' of the circle if AB = 8.

Learn how to calculate the radius of a circle using the tangent function and a given angle. This example covers trigonometric concepts and properties of right triangles, suitable for students in Grades 9-11.

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