# Find remaining vertices of a square, given 2

I need a hint for this problem.

Let the vertices of a square ABCD represent on the Argand diagram the complex numbers a,b,c, and d respectively. A,B,C,D are taken anti-clockwise in the order named.

If $$a = 3 + i, b = 4 - 2i$$, find c and d.

For a different problem, where square was at the origin, I used the idea that $$i(z_1)$$ is an anti-clockwise rotation and since it's a square etc. But here it's not at the orgin. Any help is much appreciated. Thanks.

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## 4 Answers

You can make a translation so that one of the vertices goes to the origin, then make the inverse translation.

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Got it, I did a translation of (3+i) and (4-2i) to get 2 points and then moved them back after rotation to get, C = (7-i) and D = 6 + 2i, which checks out. Thanks! – mathguy80 May 25 '11 at 13:43

Hint: $b$ to $c$ is a rotation through $\pi/2$ of $a$ to $b$.

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Hint: You can find the side of the square from $|b-a|$. If you think of walking around the square anti-clockwise, at each corner you make a left turn of $\frac{\pi}{2}$ and go the same distance.

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Haven't studied polar form of complex numbers yet, this and @Ben Bosisel idea seems more elegant one I have studied that! Until then that translation solution worked fine. Thanks. – mathguy80 May 25 '11 at 13:45
You don't need polar form, all you need to know is that when you multiply a complex number by $i$ you rotate it by 90 degrees, counterclockwise. – Gerry Myerson May 26 '11 at 0:29

A rotation in the complex plane around point $b$ with angle $\theta$ has the expression $z \mapsto b+(z-b)\cdot e^{i\theta}$, where $e^{i\theta}=\cos \theta+i\sin \theta$.

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Thanks, need to study the trignometry form of complex numbers first! – mathguy80 May 25 '11 at 13:50
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