Find the coordinates of center for the composition of two rotations

Question

The combination of a clockwise rotation about $(0, 0)$ by $120◦$ followed by a clockwise rotation about $(4, 0)$ by $60◦$ is a rotation. Find the coordinates of its center and its angle of rotation.

Here is my work so far:

$120◦+60◦=180◦$ which is not a multiple of $360◦$. As a result, the composition of these two rotations is another rotation $R_{x,\alpha}$ with the center $x$ and angle of rotation $\alpha$.

To find $\alpha$ then I divide each angle by 2, add them and multiply by 2. Therefore, $\alpha=180$

So my angle of rotation is 180 and we have $R_{x,180}$

Meanwhile, I am having a hard time finding the coordinates of its center $x$, can anyone guide me?

Answer 1

HINT: Since $\alpha=180^\circ$, the centre of the rotation must be the midpoint of any line segment joining a point to its image under the rotation. For instance, if $O$ is the origin, and $P$ is where it ends up after the composite rotation, the centre must be at the midpoint of $\overline{OP}$. It’s not hard to calculate $P$ and then find the midpoint of $\overline{OP}$.

Answer 2

What you wrote about the angle is a bit strange, you are saying that $a/2+b/2=(a+b)/2$... It's better to say something like , what would the tranformation look like far far away from these two points...

Of course this assumes that you know this theorem that a composition of two rotations is always a rotation. However, many books do teach that first, since you can characterize a rotation by what it does on arbitrary segments. Then the composition of two rotations behaves similarly, etc...

Now, for the center you just have to find the fixed point of the composition. This point should be reflected across the segment $(0,0)-(4,0)$ because of the first rotation and then back to its starting place by the second. This should be enough to tell you the angles that it defines with respect to $(0,0)-(4,0)$ and hence its coordinates...

A more straightforward way is to find the action on two particular points $A, B$. Now if $A$ is mapped to $A'$ and $B$ to $B'$, then the center should be the intersection of the perpendicular bisectors of $AA'$ and $BB'$.

Now a good choice for $A$ and $B$ involves making the actio of the composition easier... That is I would definitely have $A=(0,0)$ since the first rotation fixes it... What would be a good choice for $B$?