Which Triangle?

Geometry Level 4

If $\theta$ is real and ${ z }_{ 1 }$ , ${ z }_{ 2 }$ are connected by ${ z }_{ 1 }^{ 2 }+{ z }_{ 2 }^{ 2 }+2{ z }_{ 1 }{ z }_{ 2 }\cos { \theta } =0$ , what is the triangle with vertices $0$ , ${ z }_{ 1 }$ and ${ z }_{ 2 }$ ?

equilateral isosceles none of these right angled

1 solution

G J
Feb 20, 2021

From $z_2^{2}+2(z_2\cos\theta)z_1+z_1^{2}=0$ , we can find that $z_2=z_1\cos\theta + i z_1\sin\theta$ , hence $|z_2|=|z_1||\cos\theta+i\sin\theta|=|z_1|$ . Therefore, the triangle with vertices $0, z_1$ and $z_2$ is an isosceles triangle.

Which Triangle?

1 solution

0 pending reports