Question

Jack is on one side of a 200 foot-wide canyon and Jill is on the other side. Jack and Jill can both see the trail guide at an angle of depression of 60 degrees. How far are they from the trail guide?

Answer 1

Answer:

Jack and Jill would each be $200\; {\rm ft}$ away from the trail guide.

Step-by-step explanation:

Refer to the diagram attached. Let $\sf A$ denote Jack, $\sf I$ denote Jill, and $\sf G$ denote the trail guide.

The angle $\angle {\sf GAI} = 60^{\circ}$ as it represents the depression angle of the trail guide from the perspective Jack. Likewise, $\angle {\sf GIA} = 60^{\circ}$ for the angle from the perspective of Jill.

Note that in triangle $\triangle {\sf AGI}$:

$\begin{aligned}\angle {\sf AGI} &= 180^{\circ} - \angle {\sf GAI} - \angle {\sf GIA}\\ &= 180^{\circ} - 60^{\circ} - 60^{\circ} \\ &= 60^{\circ} \end{aligned}$.

Thus, triangle $\triangle {\sf AGI}$ would be an equilateral triangle as all its interior angles are  $60^{\circ}$. Therefore, the length of all three sides would be equal:

$\overline {\sf AG} = \overline{\sf GI} = \overline{\sf AI}$.

The length of segment $\sf{AI}$ represents the distance between Jack and Jill. This distance is supposed to be the same as the width of the canyon,  $200\; {\rm ft}$. Thus, $\overline{\sf AI} = 200\; {\rm ft}$.

The distance between Jack and the trail guide would be the same as the length of segment ${\sf AG}$, which is $\overline{\sf{AG}} = \overline{\sf AI} = 200\; {\rm ft}$.

Similarly, the distance between Jill and the trail guide would be the same as the length of segment ${\sf GI}$, which is $\overline{\sf{GI}} = \overline{\sf AI} = 200\; {\rm ft}$.