Text of the problem:
A circular loop of radius $R$ carries a current $I_1$. Perpendicular to the plane of the coil, and tangent to it, there is an indefinite rectilinear wire, traversed by a current $I_2$. Calculate the magnetic field $B$ at the centre of the coil.
The data of the problem are $R,I_1,I_2$.
So I understand the fact that the magnetic field of the wire is taken by the Biot-savart law and the magnetic field of the loop is $\mu_0i\over2R$, but in the solution (that is not well-explained) they use the Pythagorean theorem ($\sqrt{ B_1^2+B_2^2}$) to calculate the Magnetic field at the centre of the loop.
Here is a simple diagram of how the scenario is suggested:
Where the the $\color{red}{\bigcirc}$ represent the direction of current flow in the rectilinear wire. Beforehand, $$\bigcirc = \text{direction out of the diagram} \\ \times= \text{direction into the diagram}$$
I assume that I have predicted the directions of the magnetic fields correctly, using the direction of current. Nonetheless, they do not affect calculations of the magnitude of combined magnetic field.
The combined magnetic field at centre of the coil is the magnetic field generated by the coil and the magnetic field at point $Q$ combined.
If we were to rotate to the plane of the rectilinear wire, we can see that magnetic field vectors are at $90º$ to each other:
To calculate the combined magnitude we shift one vector to the head of the other, and hence we can apply the pythagorean theorem.
Hence, we have $$\color{blue}{||B_T||} = \sqrt{\color{red}{||B||^2} + \color{black}{||B||^2}}$$