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

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

51 ft.
51 ft.

120 ft.
120 ft.

111 ft.
111 ft.

20 ft.
20 ft.

GPT-4o mini · Answer

To find the length of $ LF $ in the given configuration, we can use the properties of tangents to a circle. 1. Since $ KM $ and $ KL $ are tangent to the circle at points $ M $ and $ L $, it follows that the radius $ LF $ is perpendicular to tangent $ KL $ at point $ L $ and to tangent $ KM $ at point $ M $. This forms right triangles involving point $ F $, the points of tangency, and point $ K $. 2. In triangle $ KLF $, we know: - $ \angle LFK = 23^\circ $ - $ KM = 47 \, \text{ft} $ (this is the distance from $ K $ to the point of tangency $ M $) 3. Using the tangent lengths: - Since both tangent segments from a single point outside the circle are equal, we have $ KL = KM = 47 \, \text{ft} $. 4. Now we apply the tangent and angle properties. In triangle $ KLF $: - Using the tangent of an angle: $$ \tan(\angle LFK) = \frac{LF}{KL} $$ We rearrange to find $ LF $: $$ LF = KL \cdot \tan(\angle LFK) $$ 5. Substitute the known values: - $ KL = 47 $ ft - $ \angle LFK = 23^\circ $ $$ LF = 47 \cdot \tan(23^\circ) $$ 6. Calculate $ \tan(23^\circ) $: $$ \tan(23^\circ) \approx 0.4245 $$ Then: $$ LF \approx 47 \cdot 0.4245 \approx 19.91 \, \text{ft} $$ 7. Rounding $ LF $ to the nearest foot gives: $$ LF \approx 20 \, \text{ft} $$ Thus, the length of $ LF $ is $ \boxed{20 \text{ ft}} $.