Identify and use the relationship between the radius and a line tangent to the circle, where the radius intersects the circle, to solve the following. 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?

Question

(1 point)
Responses

111 ft.
111 ft.

20 ft.
20 ft.

51 ft.
51 ft.

120 ft.
120 ft.

Answer 1

To solve for the length of \( LF \) given the tangents \( KM \) and \( KL \) to the circle, we can use the following properties of circles:

1. A radius drawn to the point of tangency is perpendicular to the tangent line at that point.
2. Since \( KM \) and \( KL \) are both tangents to the circle from point \( K \), and they are equal in length, we can express that \( KL = KM = 47 \) feet.

Given the problem provides \( \angle LFK = 23^\circ \):

In triangle \( LFK \), we have:

• The angle \( \angle LFK = 23^\circ \)
• The length of \( KL = 47 \) feet, which is one side of the triangle.

We can find the length \( LF \) using trigonometry. Specifically, we can use the tangent function, which relates the lengths of the opposite side and adjacent side in a right triangle:

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

Therefore, we can rearrange the formula to find \( LF \):

\[ LF = KL \cdot \tan(\angle LFK) \]

Now, substituting \( KL \) and \( \angle LFK \):

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

Calculating \( \tan(23^\circ) \):

Using a calculator, we find:

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

Now plugging this value into our equation:

\[ LF \approx 47 \cdot 0.4245 \approx 19.925 \]

Rounding to the nearest foot, we have:

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

Thus, the answer is:

20 ft.