Question

If a distance of 75 yds is measured back from the edge of the canyon and two angles are measured , find the distance across the canyon

Angle ACB = 50°
Angle ABC=100°
a=75 yds

What does C equal?

Answer 1

Answer:

Length of side $c$:

$c = \rm AB \approx 115\; yds$.

The distance across the canyon is approximately

$\rm 113\;yds$.

Step-by-step explanation:

• $c$ is the length of the side opposite to the angle $\rm A\hat{C} B$.
• $a$ is the length of the side opposite to the angle $\rm B\hat{A}C$.

Apply the law of sine:

$\displaystyle \frac{c}{\sin{\rm A\hat{C}B} = \frac{a}{\sin{\rm B\hat{A}C}}$.

In other words,

$\displaystyle c = \sin{\rm B\hat{A}C} \cdot \frac{a}{\sin{\rm A\hat{C} B}}$.

However, the value of the angle $\rm B\hat{A}C$ isn't given. Don't panic. The three interior angles of a triangle shall add up to 180°. Two of the three angles are given. The value of the third angle is implied.

$\rm B\hat{A}C = 180\textdegree{} - A\hat{C}B - A\hat{B}C = 30\textdegree{}$.

Apply the law of sine to find $c$:

$\displaystyle \begin{aligned}c &= \sin{\rm A\hat{C}B} \cdot \frac{a}{\sin{\rm B\hat{A}C}}\\ &= \sin{50\textdegree{}}\cdot \frac{\rm 75\; yds}{\sin{30\textdegree{}}}\\ &\rm = 114.907\; yds\end{aligned}$.

Refer to the diagram. The distance across the canyon will be

$\rm AB \cdot \sin{A\hat{B}C} = 114.907\times \sin{100\textdegree{}} \approx 113\;yds$.