In the given problem, we have the following elements:
- KM and KL are tangents to the circle.
- Angle MFK = 31 °.
- LK = 24 ft (which is one of the tangents).
Since KM and KL are tangents from point K, we know that triangle KMF is formed by line segments of the tangents and the angle between them.
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Identifying the triangle: In triangle KMF, by the properties of tangents:
- \( KM = KL \) since they are both tangents from point K to the circle.
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Finding MF: Using the tangent-secant theorem or the properties of right triangles here requires knowing some angle relationships. However, we have the angle MFK, and since you have two tangent segments from a point, you can also utilize the tangent-tangent angle theorem (the angle between two tangents coming from one point is equal to the angle between the line connecting that point to the center of the circle and the line connecting one tangent point to the other really).
For the triangle KMF:
- Let MF = x.
- Using the trigonometric tangent ratios with angle MFK (31 degrees), we know that:
\[ \tan(\angle MFK) = \frac{opposite}{adjacent} = \frac{MF}{LK} \]
So:
\[ \tan(31^\circ) = \frac{x}{24} \]
- Calculating x:
We know \( \tan(31^\circ) \approx 0.6018 \).
So, plugging in this value, we have:
\[ 0.6018 = \frac{x}{24} \]
Which can be rewritten to find x (MF):
\[ x = 24 \times 0.6018 \approx 14.44 \]
Rounding this to the nearest foot, we get:
\[ MF \approx 14 \text{ ft} \]
Thus, the answer is:
14 ft.