Complex numbers question (1 Viewer)

[pzeait]

On an argand diagram A, B, C and D represent z1, z2, z3 and z4 respectively.

i) Show that if z1-z2+z3-z4=0, then ABCD is a parallelogram

ii) Show that z1 + iz2 - z3 -iz4 = 0, then ABCD is a square.

Can I have reasoning behind all steps please? THanks

 

[integral95]

Lol I remember someone else asked the same question last year.

i) Rearrange the equation to get z1-z2 = z4-z3 which implies AB and DC are parallel and equal in length.

Rearrange again to get z1-z4 = z2-z3, meaning AD and DC are also parallel to each other and equal.

This makes ABCD a parallelogram (opposites sides are parallel and equal)

ii)it's also a parallelogram

Rearrange to get z1-z3 = i(z4-z2)

this means AC and BD cross at right angles (multiplying a vector by i rotates it by 90 degrees)

therefore ABCD is a square.


EDIT:I also have to mention the the length of AC and BD are equal, otherwise it could just be a rhombus.

 

[pzeait]

Thanks!! do you have any other examples of questions like this that i could work from?

 

[integral95]

Thanks!! do you have any other examples of questions like this that i could work from?

Not that I know of sorry.