# Geometric Proof Of Multiplication Rule Activity

- Author:
- Gary Rubinstein

- Topic:
- Multiplication

C and B represent two complex numbers. H is the product of C and B. The goal of this activity is to prove that the magnitude of H is equal to the product of the magnitude of C and the magnitude of B and that angle of H (with respect to the positive x-axis) is equal to the sum of the angles for C and B.

Drag points B and C around to see if you notice anything about how all the other points and segments in the diagram are affected by B and C. Click on the 'refresh' icon on the top right of the sketch to restore the points to the original B=3+4i and C=1+5i

If C=a+bi and B=c+di, then:
E=(a+bi)*c
F=(a+bi)*i
G=(a+bi)*di
H=E+G=(a+bi)*c+(a+bi)*di=(a+bi)(c+di)=C*B

## Question 1

Explain why triangle AHE must be similar to triangle ABD

## Question 2

Explain why the fact that those triangles are similar means that angle DAH must equal the sum of angles DAB and DAE.

## Question 3

Explain why the fact that those triangles are similar means that the length of OH must be the product of the lengths of AB and AE.