Complex numbers

Not sure how to translate the point (2,3) back to the origin

What have you done so far? Have done the cube roots of "unity" then rotation and translation? Or ...

I've worked out the roots of unity for triangle centre about origin, but can't seem to figure how to translate the origin to (2,3) so that I can apply it to the other roots

I've worked out the roots of unity for triangle centre about origin, but can't seem to figure how to translate the origin to (2,3) so that I can apply it to the other roots

If $z^3 = 1$ provides the cubes of unity about the origin, then $(z-z_0)^3 = 1$ provides the cubes of unity about the point $z_0$. Set it appropriately to get the centre at (2,3) on the complex plane.

But the easier approach would just be to see what is the complex number (i.e. vector) that takes you from (2,3) to (3,-2) and then just rotate it 120 degrees by multiplication by an appropriate complex number. Then add the rotated vector onto the center in order to go straight to where another vertex is.

Add (2,3) to each root. Im assuming that youve scaled by 26^1/2 and rotated by the correct arg first.

Note once you know the arg rotation, you could almost write down the correct trig expressions without doing any complex stuff.

Having drawn a simple sketch I realised that the transformations compromises as enlargement and rotation then addition, however I'm a bit unsure as to how you would find the rotation, Is it essentially just finding the angle difference between e^(0i) and the vertex given in the question then using modulus argument form you can correct the orientation then enlarge before adding the complex number 2+3i

You know one vertex, relative to the center is (1,-5) which is arg atan(-5). So if you rotate the x-axis "root" by that amount, then all three "roots" would be rotated by that. Or alternatively add 2pi/3 and 4pi/3 to that angle and forget about the complex dressing and just use simple trig.

This what I've done which is what I think you mean

a = atan(-5), r = sqrt(26), c=(2,3)
z1 = c + r*e^(ai)
z2 = c + r*e^((a+2pi/3)i)
z3 = c + r*e^((a+4pi/3)i)
Tbh, there is little complex numbers here, Im just using it to avoid writing cis (cos(),sin())

Is this the correct approach, are there any other methods?