Question

1. problem solving a boat is traveling parallel to the shore along \\(\overrightarrow{rt}\\). when the boat is at point \\(r\\), the captain measures the angle to the lighthouse as \\(35^\circ\\). after the boat has traveled 2.1 miles, the captain measures the angle to the lighthouse to be \\(70^\circ\\).

image of a diagram with points r, s, t on a horizontal line (\\(\overrightarrow{rt}\\)) with rs = 2.1 mi, angle at r (\\(\angle lrs\\)) is \\(35^\circ\\), angle at s (\\(\angle lst\\)) is \\(70^\circ\\), and point l below the line representing the lighthouse.
a. find \\(sl\\). explain your reasoning.
b. explain how to find the distance between the boat and the shoreline.

Explanation:

Step1: Analyze Triangle Angles

In $\triangle RSL$, $\angle R = 35^\circ$, $\angle LST = 70^\circ$. By exterior angle theorem, $\angle LST=\angle R+\angle RLS$. So $\angle RLS = 70^\circ - 35^\circ = 35^\circ$. Thus, $\triangle RSL$ is isosceles with $\angle R=\angle RLS$.

Step2: Apply Isosceles Triangle Property

In an isosceles triangle, sides opposite equal angles are equal. So $SL = RS$. Given $RS = 2.1$ miles, hence $SL = 2.1$ miles.

Answer:

$SL$ is 2.1 miles.