1. ## Question

As shown in the diagram, a cottage is located at C on an island in the lake, and a marine is located at B. If the distance from A to D is 10.0 km, and <ABC = <CAB = 28˚, find the distance from B to C.

2. Hello, Ph4m!

As shown in the diagram, a cottage is located at C on an island in the lake,
and a marine is located at B.
If the distance AD = 10.0 km, and <ABC = <CAB = 28°, find distance BC.
Code:
                              A
*
* **
*   * *
* 28° *  *
*       *   * 10.0 km
*         *x   *
*           *     *
* 28°    124° * 56°  *
* * * * * * * * * * * * *
B       x       C       D
Let $\displaystyle x = BC$.

Since $\displaystyle \angle ABC = \angle CAB = 28^o$, triangle ABC is isosceles.
Hence: .$\displaystyle AC = BC = x$.
Since $\displaystyle \angle ACB \:= \:180^o - 28^o - 28^o \:=\:124^o$, then: .$\displaystyle \angle ACD = 56^o$

In right triangle ADC, we have: .$\displaystyle \sin56^o \:=\:\frac{10}{x}\quad\Rightarrow\quad x \:=\:\frac{10}{\sin56^o} \:=\:12.06217949$

Therefore: .$\displaystyle BC \:\approx\:12.1$ km.