Hello, <3<3!

I assume you made sketches . . .

Town $\displaystyle Y$ is 9 km due north of town $\displaystyle Z.$

Town $\displaystyle X$ is 8km from $\displaystyle Y$, 5km from $\displaystyle Z$ and somewhere to the west of the line $\displaystyle YZ.$

A) Draw $\displaystyle \Delta XYZ$ and find $\displaystyle \angle YZX$ Code:

* Y
* |
8 * |
* |
* | 9
* |
X * |
* |
5 * θ |
* |
* Z

Let $\displaystyle \theta = \angle YZX$

Law of Cosines: .$\displaystyle \cos\theta \:=\:\frac{5^2 + 9^2 - 8^2}{2(5)(9)} \:=\:0.46666...$

Therefore: .$\displaystyle \theta \:\approx\:62.2^o$

B) During an earthquake, town $\displaystyle X$ moves due south until it is due west of $\displaystyle Z.$

Find how far it has moved. Code:

* Y
* |
8 * |
* |
* | 9
* |
X * 5 |
: * 62.2°|
: * |
: 27.8° * |
W * - - - - - * Z

Town $\displaystyle X$ has moved directly south to $\displaystyle W.$

Since $\displaystyle \angle YZX = 62.2^o$, then: .$\displaystyle \angle XZW = 27.8^o$

In right triangle $\displaystyle XWZ\!:\;\;\sin27.8^o \:=\:\frac{XW}{5} \quad\Rightarrow\quad XW \:=\:5\sin27.8^o \:=\:2.3319...$

Therefore, town $\displaystyle X$ moved about 2.3 km.