identify and use the relationship between the radius and a line tamgent to the circle, where the radius intersects the circle, to solve the folowing. if KM and KL are tangent to the circle at L and M, <LFK=23 degrees and KM= 47 feet, how long is LF to the nearest foot.

Answer 1

To solve this problem, we can use the properties of circles, particularly the relationship between a radius and a tangent line at the point of tangency.

1. Identify the Circle and the Tangents:

   • Let \( O \) be the center of the circle.
   • \( K \) is a point outside the circle, and \( KM \) and \( KL \) are tangent lines to the circle at points \( M \) and \( L \), respectively.
   • By the properties of tangents, the radius \( OL \) is perpendicular to the tangent line \( KL \) at point \( L \).

2. Use of the Right Triangle:

   • \( \triangle LFK \) is formed by the radius \( OL \) and the tangent \( KL \).
   • Given that \( \angle LFK = 23^\circ \) and the tangent length \( KM = 47 \) feet, we can determine the length of \( LF \).

3. Use of Trigonometry:

   • In the right triangle \( \triangle LFK \):
     • \( LF \) is the opposite side to the angle \( \angle LFK \).
     • \( KL \) is the adjacent side, which we need to find.

Using the tangent function since we know the angle and one side:

\[ \tan(\angle LFK) = \frac{\text{opposite}}{\text{adjacent}} = \frac{LF}{KL} \]

We already know the relationship between the tangents:

\[ KL = KM = 47 \text{ feet} \]

Thus,

\[ \tan(23^\circ) = \frac{LF}{47} \]

Now, calculate \( LF \):

\[ LF = 47 \cdot \tan(23^\circ) \]

4. Calculating \( LF \):

Using a calculator to find \( \tan(23^\circ) \):

\[ \tan(23^\circ) \approx 0.4245 \]

Now calculating \( LF \):

\[ LF \approx 47 \cdot 0.4245 \approx 19.93 \]

Rounding to the nearest foot, we find:

\[ LF \approx 20 \text{ feet} \]

Thus, the length of \( LF \) is approximately 20 feet.