Question

A vertical line connects upper K to upper F. Diagonal lines connect upper K with upper L and upper K with upper M. Similarly, the points upper L and upper F are connected via a diagonal line and the points upper M and upper F are connected via a diagonal line.

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

20 ft.
20 ft.

111 ft.
111 ft.

51 ft.
51 ft.

120 ft.

Answer 1

Because a radius to a tangent point is perpendicular to the tangent, triangle FKL is right at L. Let r = FL and d = FK. Then KL = 47, and in the right triangle

sin(23°) = KL / d = 47 / d => d = 47 / sin(23°).

Also cos(23°) = r / d, so r = d cos(23°) = 47 cot(23°) ≈ 110.66 ft.

Rounded to the nearest foot: 111 ft.