To solve the problem, we need to analyze the relationship between the circle's properties and the given information about the tangent lines.
-
Understanding the geometry of the situation:
- We have a circle with center \( F \).
- Points \( L \) and \( M \) are where the tangents \( KL \) and \( KM \) touch the circle.
- The angles created by the radius and the tangent at the point of tangency prominently help us solve for the radius.
-
Using known properties:
- The radius \( FL \) is perpendicular to the tangent \( KL \) at point \( L \).
- Hence, we form a right triangle \( FKL \) where:
- \( FK \) is the distance from the center \( F \) to point \( K \).
- \( FL \) is the radius we are trying to find.
- \( KL \) is the length of the tangent from point \( K \) to point \( L \).
-
Identifying triangle properties:
- In triangle \( FKL \), we know:
- \( KL = 47 , \text{ft} \) (the length of the tangent).
- The angle \( ∠LFK = 23^\circ \).
- In triangle \( FKL \), we know:
-
Using trigonometric relationships:
- By applying the tangent of the angle: \[ \tan(∠LFK) = \frac{opposite}{adjacent} = \frac{FL}{KL} \]
- Rewrite this in terms of \( FL \): \[ \tan(23^\circ) = \frac{FL}{47} \]
-
Solving for \( FL \):
- Calculate \( FL \): \[ FL = 47 \times \tan(23^\circ) \]
-
Calculating \( \tan(23^\circ) \):
- Using a calculator or trigonometric tables: \[ \tan(23^\circ) \approx 0.4245 \]
- Therefore: \[ FL \approx 47 \times 0.4245 \approx 19.94 , \text{ft} \]
-
Rounding to the nearest foot:
- \( FL \approx 20 , \text{ft} \).
The answer is 20 ft.