To find the length of \( LF \), we first need to understand the geometric relationships in the problem. Given the conditions:
- \( KM \) and \( KL \) are tangent to a circle at points \( M \) and \( L \), respectively.
- \( \angle LFK = 23^\circ \)
- \( KM = 47 , \text{ft} \)
Since \( KM \) and \( KL \) are tangents to the circle, and \( K \) is the point of intersection of the tangents, \( \triangle KML \) is a right triangle, where the angles at \( M \) and \( L \) are right angles.
We can apply the following relationships:
- The tangents from a point outside the circle (in this case, point \( K \)) to the circle are equal in length. Thus, \( KM = KL = 47 , \text{ft} \).
- In triangle \( KLF \):
- \( KL \) is adjacent to angle \( LFK \).
- \( LF \) is opposite to angle \( LFK \).
Using the tangent of angle \( LFK \): \[ \tan(\angle LFK) = \frac{\text{opposite}}{\text{adjacent}} = \frac{LF}{KL} \] Thus, \[ \tan(23^\circ) = \frac{LF}{47} \]
Now we solve for \( LF \): \[ LF = 47 \cdot \tan(23^\circ) \]
Next, we calculate \( \tan(23^\circ) \): Using a calculator, \[ \tan(23^\circ) \approx 0.4245 \]
Now substituting this value back into our equation for \( LF \): \[ LF = 47 \cdot 0.4245 \approx 19.9125 \]
Rounding to the nearest foot: \[ LF \approx 20 , \text{ft} \]
Thus, the length of \( LF \) is approximately 20 ft.
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