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=\frac{15}2`