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Math Help - Complex Numbers - Triangles

  1. #1
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    Complex Numbers - Triangles

    Let z be a complex number with |z| = 6. Let A be the point representing z. Let B be the point representing (1 + i)z.
    a)
    i. Find |(1 + i)z|
    . Done. The answer is 6\sqrt{2}.
    ii. Find |(1 + i)z - z|. Done. The answer is 6.
    iii. Prove that OAB is an isosceles right-angled triangle. Is this just a matter of a2 + b2 = c2?
    b) Let z1 and z2 be non-zero complex numbers satisfying z12 - 2z1z2 + 2z22 = 0.
    If z1 = $\alpha$ z2:

    i) show that $\alpha$ = 1 + i or 1 - i. Done.
    ii) for each of these values of $\alpha$ describe the geometrical nature of the triangle whose vertices are the origin and the points representing z1 and z2​. ​How would I go about doing this?
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  2. #2
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    Re: Complex Numbers - Triangles

    Hi,
    I hope the attachment answers your geometric questions.

    Complex Numbers - Triangles-mhfgeometry64.png
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  3. #3
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    Re: Complex Numbers - Triangles

    a) i) In complex numbers, each complex number works the same way as a vector. Segment OA is z. Segment OB is (1 + i)z. Using vector algebra, that means AB = OB - OA = (1 + i)z - z, thus they form three sides of a triangle. Because the three sides are in the ratio $\displaystyle \begin{align*} 6 : 6 : 6\sqrt{2} = 1 : 1 : \sqrt{2} \end{align*}$, the triangle is isosceles right angled.
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