Question

I dont really know how to do soh cah toah. please help me find x.

Answer 1

First notice that the triangle with sides $x,y,a$ and the triangle with sides $z,a+b,x$ are similar. This is true because the angle between sides $x,y$ in the smaller triangle is clearly $30^\circ$, while the angle between sides $y,z$ in the larger triangle is clearly $60^\circ$. So the triangles are similar with sides $x,y,a$ corresponding to $a+b,z,x$, respectively.

Now both triangles are $30^\circ-60^\circ-90^\circ$, which means there's a convenient ratio between its sides. If the length of the shortest leg is $\ell_1$, then the length of the longer leg is $\ell_2=\sqrt3\ell_1$ and the hypotenuse has length $\ell_3=\sqrt{{\ell_1}^2+{\ell_2}^2}=2\ell_1$.

Since $x$ is the shortest leg in the larger triangle, it follows that $a+b=2x$, so $a+b=15=2x\implies x=\dfrac{15}2$