A circle has center point upper F. The circumference point upper L is at the left of the circle and the circumference point upper M is at the right of the circle. Point upper K is above the circle in line with upper F.
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?
(1 point)
Responses
111 ft.
111 ft.
20 ft.
20 ft.
51 ft.
51 ft.
120 ft.
51 ft.
To solve this problem, we can use the relationship between the radius of a circle and a tangent line that intersects the circle. The radius of a circle is perpendicular to a tangent line at the point of tangency. This means that triangles FKL and FKM are right triangles.
We are given that angle LFK is 23 degrees and that KM is 47 feet. Since triangles FKL and FKM are right triangles, we can use trigonometry to find LF.
In triangle FKL, tan23 = LF/KL. Since KL is the radius of the circle, KL = LF. So, tan23 = LF/LF.
Therefore, LF = 47/tan23 ≈ 51 feet.
Therefore, LF is approximately 51 feet.