Area of a Parallelogram Graphed on Real and Complex Plane

Algebra Level 3

The Argand Diagram is a method of graphing points similar to the Cartesian coordinate system. Instead, however, points on the Argand Diagram, also called the complex coordinate plane, are represented as complex numbers in the form $a+bi$ , where the real component $a$ represents the $x$ -coordinate, and the imaginary component $b$ represents the $y$ -coordinate. When $f(x)=-x^{2}+6x-45$ is graphed on the Cartesian coordinate plane, let the two roots be $z_1$ and $z_2$ . The points $(8, 13)$ and $(8, 1)$ are graphed on the Cartesian coordinate plane, and $z_1$ and $z_2$ are graphed on the Argand Diagram. Then, they two systems are overlapped such that their origins are equal and they are proportionate (i.e. the point $3i$ on the Argand Diagram is equal $(0, 3)$ on the Cartesian coordinate plane). What is the area of the parallelogram formed by the four points?

The answer is 60.

1 solution

Finn Hulse
Apr 9, 2014

First, solve for $z_1$ and $z_2$ . For the sake of brevity, let's use the quadratic formula: $z_1, z_2=\frac{-6 \pm 12i}{-2} \Longrightarrow 3 \pm 6i$ . Now, graph these points as if they are Cartesian. This produces $(3, 6)$ and $(3, -6)$ . The other two points given are $(8, 13)$ and $(8, 1)$ . Forming the parallelogram, we see that it has a height of five and a base length of 12. Thus, $\boxed{60}$ is the parallelogram's area.

Area of a Parallelogram Graphed on Real and Complex Plane

The answer is 60.

1 solution

0 pending reports