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?
First, we need to use the properties of a circle and a tangent line to determine that angle LKM is a right angle. This is because a tangent line is perpendicular to the radius of a circle at the point of tangency.
Since angle LKM is a right angle, angle LFK is complementary to angle LKM. Therefore, angle LFK is 90 degrees - 23 degrees = 67 degrees.
Now we can use trigonometry to find the length of LF. In right triangle LKF, we have the opposite side (LF), the adjacent side (FK), and the angle LFK.
Using the tangent function:
tan(67 degrees) = LF / FK
tan(67 degrees) = LF / 47 feet
Solving for LF:
LF = tan(67 degrees) * 47 feet
LF ≈ 112 feet
Therefore, LF¯¯¯¯¯¯¯ is approximately 112 feet long.