Question

If the area of the triangle formed formed `z`, `wz` and `z=wz` where `w` is the cube root of unity is `16sqrt3 `sq units then `|z|=`

Answer 1

Please refer to video to see the diagram.
As `omega` is the cube root of unity,
So, angle between `omega` and `omegaz` will be `120^@`.
As, sum of two vertices of the traingles `omega` and `omegaz` is equal to `omega+omegaz`,
so, the quadrilateral in the diagram is a parallelogram.
So, the remaining angles of the traingle will be `30^@`.
Now, area of the triangle, `A = 1/2*b*h`
`16sqrt3 = 1/2**2|z|cos30^@**|z|sin30^@`
`=>16sqrt3 = 1/2|z|^2sin60^@`
`=>32sqrt3 = |z|^2*sqrt3/2`
`=>64 = |z|^2`
`=>|z| = 8.`