Solving Complex Number Problems: Find b and d Using an Argand Diagram

[coffeebeans]

Hi,
I desperately need help with this qns:
In an Argan Diagram, the points A, B, C, D represent the copmlex numbers a,b,c,d respectively. Guiven that ABCD is a rectangle describd in an anticlocwise sense, with AB=2CB, and a=-2-i, c=3+5i, find b and d

Any help is greatly appreciated, thnx loads!

 

[roger]

Just draw a picture if your not sure and then it should be easy to see

 

[coffeebeans]

I have! I've drawn the diagramme but i carn solve it and neither can my dad. I;m sorry, can u help me please? I;m doing Alvl pure maths (not further) by the way. thnx!

 

[roger]

so if you plot the points, b lies somewhere on x=3 and d somewhere on x=-2

and all the lines are perpendicular, so you can work out the i values of the point b and d

 

[coffeebeans]

nono, but AB doesn't have to be horizontal (i.e. parallel to the x axis) the rectangle can and probably would be slanted. the condition given is AB=2CD. if AB and CD are parallel to the xaxis the condition is not satisfied

 

[VietDao29]

So: you have AB = 2BC \Rightarrow AB ^ 2 = 4 BC ^ 2
ABC is the right triangle. You also have the length of AC. What does this suggest you?
Let b = x_B + iy_B
So B(x_B, y_B)
You also have \overrightarrow{BA}\overrightarrow{BC} = 0
So you will come up with 2 equations:
\left\{ \begin{array}{l} (x_B + 2)(y_B - 3) = -(y_B + 1)(y_B - 5) \\ AB ^ 2 = (x_B + 2) ^ 2 + (y_B + 1) ^ 2 = ... \end{array} \right.
From there you can solve for b, then d.
Viet Dao,

 

[coffeebeans]

i did tht b4 i posted the qns. Tht;s where i got stuck cos the numbers were absurdly big and unfriendly, but its ok, I figured another way of doing already. thnx for ur help anyway!