Question

evelyn watches airplanes take off at an airport. she is 200 meters away from where the planes take off from the runway. the initial angle at takeoff of a plane is 12°. what is true when the plane is 100 meters away from takeoff? check all that apply. the given information is the sss case. the given information is the sas case. the given information is the ssa case. evelyn is approximately 104 meters from the plane. the angle of elevation of the plane is 12°. (image of a triangle with labels: 100 m, 200 m, 12°, and evelyn at one vertex, the plane, and the takeoff point at another vertex)

Explanation:

Step1: Recall triangle - case definitions

• SSS (Side - Side - Side): three side lengths are known.
• SAS (Side - Angle - Side): two sides and the included angle are known.
• SSA (Side - Side - Angle): two sides and a non - included angle are known.

Here, we have two sides (\(a = 100\), \(b = 200\)) and a non - included angle (\(A=12^{\circ}\)), so it is the SSA case.

Step2: Use the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In our case, if we want to find the distance \(d\) between Evelyn and the plane (let \(a = 100\), \(b = 200\), \(C = 12^{\circ}\)), then \(d^{2}=100^{2}+200^{2}-2\times100\times200\times\cos(12^{\circ})\).
\(\cos(12^{\circ})\approx0.9781\)
\(d^{2}=10000 + 40000-40000\times0.9781\)
\(d^{2}=50000-39124\)
\(d^{2}=10876\)
\(d=\sqrt{10876}\approx104\)

Answer:

The given information is the SSA case. Evelyn is approximately 104 meters from the plane.