1. ## Complex Geometry

The points A and B in the complex plane correspond to complex nos. z1 and z2 respectively. Both triangle OAP and OBQ are right angled isosceles triangles.

i) Explain why P correspond to (1+i)z1: Done
ii) M is midpoint of PQ, what complex no. correspond to M

Here is my attempt, which conflict with solutions

Q reqpresents z2(1-i), by rotation.

Since M is mid point of QP, then QM = 1/2 (p-q)

=1/2 (z1(1+i)-z2(1-i))
...
=1/2(z1-z2) + i/2 (z1+ z2)

But QM = M - z2
z2 + QM = M
therefore,
M = 1/2 ((z1+z2) + i(z1 + z2))

However solutions has 1/2(z1+z2 + i(z1-z2))

2. Originally Posted by Lukybear
The points A and B in the complex plane correspond to complex nos. z1 and z2 respectively. Both triangle OAP and OBQ are right angled isosceles triangles.

i) Explain why P correspond to (1+i)z1: Done
ii) M is midpoint of PQ, what complex no. correspond to M

Here is my attempt, which conflict with solutions

Q reqpresents z2(1-i), by rotation.
Why does Q not correspond to (1+i)z2, given that you have already accepted that P corresponds to (1+i)z1, accordingt to i)?
It seems to me that you are getting P by rotating A around 0 by +90 degrees, whereas you are getting Q by rotating B around 0 by -90 degrees. Which is right: rotating by +90 or rotating by -90 degrees?

Since M is mid point of QP, then QM = 1/2 (p-q)

=1/2 (z1(1+i)-z2(1-i))
...
=1/2(z1-z2) + i/2 (z1+ z2)

But QM = M - z2
z2 + QM = M
therefore,
M = 1/2 ((z1+z2) + i(z1 + z2))

However solutions has 1/2(z1+z2 + i(z1-z2))

3. Sorry to classify, P and Q are separate isoceles right triangles, with P in 1st quad and Q in 4th quad. So for this case lets assume my P and Q are correct. What about resulting solution?

So this is what it would look like. Sorry for confusion.

4. Originally Posted by Lukybear
Sorry to classify, P and Q are separate isoceles right triangles, with P in 1st quad and Q in 4th quad. So for this case lets assume my P and Q are correct. What about resulting solution?

So this is what it would look like. Sorry for confusion.
The midpoint M is $\displaystyle M=\frac{P+Q}{2}$ (I am treating points like complex numbers for simplicity here).

Thus you get

$\displaystyle M=\frac{P+Q}{2}=\frac{(1+i)z_1+(1-i)z2}{2}=\frac{(z_1+z_2)+i(z_1-z_2)}{2}$

You can get the same result like this
$\displaystyle M=P+\frac{1}{2}\cdot (Q-P)=(1+i)z_1+\frac{1}{2}\cdot\big((1+i)z_2-(1-i)z_1\big)=\ldots$

5. Thanks for that.