Question

nigel and mia are searching for a treasure chest under water. the straight line distance between them is 100 meters.
nigel 100 m mia
65° 80°
treasure chest
given the angles in the diagram, who is closer to the treasure chest and why?
○ nigel is closer because his distance to the chest is opposite the larger angle.
○ mia is closer because her distance to the chest is opposite the smaller angle.
○ nigel is closer because his distance from the chest is 100 meters.
○ mia is closer because her distance from the chest is 100 meters.

Explanation:

Step1: Recall the Law of Sines

In a triangle, the Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where $a, b, c$ are the lengths of the sides opposite angles $A, B, C$ respectively. This means that the side opposite a larger angle is longer, and the side opposite a smaller angle is shorter.

Step2: Analyze the angles in the triangle

In the triangle formed by Nigel, Mia, and the treasure chest, the angle at Nigel is $65^\circ$, the angle at Mia is $80^\circ$, and the side between Nigel and Mia is $100$ meters (opposite the angle at the treasure chest). Let the distance from Nigel to the treasure chest be $m$ (opposite the $80^\circ$ angle) and the distance from Mia to the treasure chest be $n$ (opposite the $65^\circ$ angle).

Since $65^\circ<80^\circ$, by the Law of Sines, the side opposite the smaller angle ($65^\circ$, which is Mia's distance to the chest) will be shorter than the side opposite the larger angle ($80^\circ$, which is Nigel's distance to the chest). So Mia's distance to the chest is shorter, meaning she is closer. The reason is that her distance to the chest is opposite the smaller angle.

Answer:

Mia is closer because her distance to the chest is opposite the smaller angle.