In the complex plane, z and ∣z∣ are vectors of equal length, and z+∣z∣=2+8i is their vector sum. So 0,z,2+8i, and ∣z∣ form the vertices of a rhombus, as in the figure. The diagonals of a rhombus bisect each other and are perpendicular. The diagonal from 0 to 2+8i has slope 4 and midpoint 1+4i. Thus the diagonal from z to ∣z∣ passes through 1+4i with slope −1/4. Therefore this diagonal intersects the real axis at x=17. Since ∣z∣ is on the real axis, we conclude that ∣z∣=17 and ∣z∣2=289.
