To solve for the length of LF, we can use the properties of tangents and some trigonometry.
Given:
- \( KM \) and \( KL \) are tangents to the circle.
- Angle \( LFK = 23^\circ \)
- Length \( KM = 47 , \text{ft} \)
Properties:
- The length of tangents drawn from an external point to a circle are equal; hence, \( KL = KM = 47 , \text{ft} \).
- Angles formed between the tangent and a line drawn to the point of tangency are equal to the angle formed by the radii.
Steps to Find LF:
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Identify Triangle:
- Triangle \( LFK \) is formed by points \( L \), \( F \), and \( K \).
- In this triangle, \( LF \) is opposite to angle \( LFK \) and \( KL \) is the hypotenuse.
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Use Trigonometric Relation:
- In triangle \( LFK \): \[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \]
- According to the values: \[ \sin(23^\circ) = \frac{LF}{KL} \]
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Substituting Known Values:
- We know \( KL = KM = 47 , \text{ft} \).
- Thus: \[ \sin(23^\circ) = \frac{LF}{47} \]
- Rearranging this gives: \[ LF = 47 \cdot \sin(23^\circ) \]
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Calculate \( LF \):
- Using a calculator, find \( \sin(23^\circ) \): \[ \sin(23^\circ) \approx 0.3907 \]
- Now calculate \( LF \): \[ LF = 47 \cdot 0.3907 \approx 18.3 , \text{ft} \]
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Rounding to the Nearest Foot:
- Rounding \( 18.3 \) to the nearest foot gives: \[ LF \approx 18 , \text{ft} \]
Final Answer:
The length of \( LF \) to the nearest foot is 18 ft.
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