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:[br]E=(a+bi)*c[br]F=(a+bi)*i[br]G=(a+bi)*di[br]H=E+G=(a+bi)*c+(a+bi)*di=(a+bi)(c+di)=C*B[br]
Explain why triangle AHE must be similar to triangle ABD
Explain why the fact that those triangles are similar means that angle DAH must equal the sum of angles DAB and DAE.
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.