Question

solving for the measures of a triangle
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 $12^{\circ}$.
what is true when the plane is 100 meters away from takeoff? check all that apply.
$square$ the given information is the sss case.
$square$ the given information is the sas case.
$square$ the given information is the ssa case.
$square$ evelyn is approximately 104 meters from the plane.
$square$ the angle of elevation of the plane is $12^{\circ}$.

Explanation:

Step1: Identify triangle case

We have two sides: 200 m (Evelyn to takeoff point), 100 m (takeoff point to plane), and the included angle \(12^\circ\) between them. This is the SAS (Side-Angle-Side) case.

Step2: Calculate distance to plane

Use the Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos(C)\), where \(a=200\), \(b=100\), \(C=12^\circ\)
\[

$$\begin{align*} c^2 &= 200^2 + 100^2 - 2(200)(100)\cos(12^\circ)\\ c^2 &= 40000 + 10000 - 40000(0.9781)\\ c^2 &= 50000 - 39124\\ c^2 &= 10876\\ c &=\sqrt{10876} \approx 104 \end{align*}$$

\]

Step3: Evaluate other options

• SSS requires 3 sides (we only have 2 initially), SSA requires 2 sides and non-included angle (not our case). The \(12^\circ\) angle is at the takeoff point, not the angle of elevation from Evelyn.

Answer:

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