To find the length of \( LF \), we can use the relationship between the radius and the tangent line. The radius at the point of tangency is perpendicular to the tangent line. In this case, we have a triangle \( LFK \) where:
- \( LF \) is the radius of the circle (from the center \( F \) to point \( L \)).
- \( KL \) and \( KM \) are tangent segments to the circle, making \( KL = KM = 47 \) feet.
- \( \angle LFK = 23^\circ \).
Since \( KL \) is tangent to the circle at point \( L \), the radius \( LF \) is perpendicular to the tangent \( KL \) at point \( L \), so \( \angle LKF = 90^\circ \).
Now we can use the definition of the tangent, which gives us a right triangle \( LFK \) with:
- \( LF \) as one side (the radius we need to find).
- \( KL = 47 \) feet as the other side (the tangent segment).
- \( \angle LFK = 23^\circ \).
To find \( LF \), we can use the tangent function:
\[ \tan(\angle LFK) = \frac{LF}{KL} \]
Substituting the known values into the equation:
\[ \tan(23^\circ) = \frac{LF}{47} \]
Now, rearranging for \( LF \):
\[ LF = 47 \tan(23^\circ) \]
Calculating \( \tan(23^\circ) \):
\[ \tan(23^\circ) \approx 0.4245 \]
Now calculate \( LF \):
\[ LF \approx 47 \times 0.4245 \approx 19.93 \]
Rounding \( LF \) to the nearest foot gives us:
\[ LF \approx 20 \text{ feet} \]
So the final answer is:
20 ft.
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