Question

Two opposite vertices of a square are given by the complex numbers z1=-3+7i and z2=9-9i. Find the other two vertices of the square and give also the equation of its circumscribed circle.

Any tips are appreciated.

Answer 1

I was thinking to find the mid point between the two vertices and then find the other two vertices using that mid point and different directions. I already found the mid point (3-i).

Answer 2

@c0rtez


Do you know rotation ??

i.e
\[\large z_4 - z_1 = \frac{z_2-z_1}{|z_2-z_1|} * |z_4-z_1|\]

Since |z_4-z_1| = |z_2-z_1| (side of square are equal )

Now you will get z_4. Similarly find z_3

Answer 3

oops . Sorry It should be
\[\large z_4 - z_1 = \frac{z_2-z_1}{|z_2-z_1|} * |z_4-z_1| * e^{i (\-pi /2)}\]

Answer 4

Typo:
e^(i * pi/2)

Answer 5

Ok thanks I will try to solve it now.

Answer 6

Sorry typo again : e^(i * -pi/2)

Answer 7

So if I got it right, I can multiply z1 by e^(i * pi/2) to get z2, and use -pi/2 for z4?

Answer 8

@c0rtez, It should be
\[\large z_4 - z_1 = \frac{z_2-z_1}{|z_2-z_1|} * |z_4-z_1| * e^{-i \pi /2}\]
Since |z_4-z_1| = |z_2-z_1| (side of square are equal )

Now you will get z_4 from above

Use the same technique and find z_3