Complex Area

Algebra Level 4

Let complex number $z = 3 + 4i$ . If the area of the triangle made by the complex numbers $z$ , $iz$ and $z+iz$ is of the form $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find the value of $a+b$ .

The answer is 27.

1 solution

Steven Chase
Jan 1, 2017

$z = 3 + 4i = \vec{v_1} \\ iz = -4 + 3i\ = \vec{v_2} \\ z + iz = -1 + 7i = \vec{v_3}$

These are the coordinates of the three triangle vertices. Calculate vectors associated with two of the triangles sides.

$\vec{v_{s1}} = \vec{v_3} - \vec{v_2} = 3 + 4i \\ \vec{v_{s2}} = \vec{v_1} - \vec{v_2} = 7 + i$

The area of the triangle is one half the magnitude of the cross product of $\vec{v_{s1}}$ and $\vec{v_{s2}}$ , which comes out to $\frac{25}{2} = \frac{a}{b}$ .

$a+b = 27$ .

It's interesting that at first glance, you would speculate that the triangle would be degenerate since the "sum" of two sides equals the third, i.e., $z + iz = z + iz$ . But since we can only order complex numbers by taking their moduli, we can only apply the triangle inequality by looking at $|z|, |iz|$ and $|z + iz|$ , which are $5,5$ and $5\sqrt{2}$ , respectively, indicating that we are dealing with an isosceles right triangle.

Brian Charlesworth - 4 years, 5 months ago

Complex Area

The answer is 27.

1 solution

0 pending reports