Relativistic Addition of Orthogonal Velocities

Stuck on a homework question and unsure how to approach orthogonal addition. Any help would be appreciated.

In a given inertial frame, two particles are shot out simultaneously from a given point,
with equal speed v, in orthogonal directions.
Using a conventional approach to solving relative velocity problems, and without using
4-vectors, show that the speed of each particle relative to the other is given by:

$$v_R = vleft(2- left(frac{v}{c}right)^{2}right)^{1/2}$$

Derive the same result using 4-velocity vectors.

What I tried:

Taken inertial frame S. Considering particles emitted in x and y directions.
Let S’ be the frame of the particle going in y direction where
$x=0$,
$y=vt$
$$x’= -?vt$$
$$y’ = vt$$

Inverse lorentz transformation from S’ to S
$$t=?(t’ + vx/c^2) $$
with $x=0$ gives
$$t=?t’$$

Substitute into $x’$ and $y’$

$$x’ = -?^2vt’$$

$$y’ = ?vt’$$

Not sure where I would go for here or if this is the right approach to take 🙂