Find a square using a quarter-turn · source reference

Question

A, B, C and D represent $-2+5i,z_1,4+i,z_2$ respectively. ABCD is a square labelled anticlockwise. Show $z_1=-1$ and find $z_2$. [4]

[2016/VJC/Prelim/II/3]

Source working

Printed working

BC represents $4+i-z_1$; BA represents $-2+5i-z_1$.

$$\arg(4+i-z_1)+\pi/2=\arg(-2+5i-z_1).$$ $$\begin{aligned}i(4+i-z_1)&=-2+5i-z_1\\4i-1-iz_1&=-2+5i-z_1\\(1-i)z_1&=-1+i\\z_1&=-1.\end{aligned}$$

The printed midpoint line is $((-2+4)/2,(5-1)/2)=(1,3)$.

Let $z_2=x+iy$. The diagonals of a square bisect each other, hence BD has midpoint $(1,3)$:

$$((x-1)/2,(y+0)/2)=(1,3)\Rightarrow x=3,y=6,$$ $$z_2=3+6i.$$

Teacher note

A is obtained when C is rotated through $90^\circ$ anticlockwise about B.

Correction of midpoint typo

The midpoint of A and C uses $(5+1)/2=3$, not the printed $(5-1)/2$. The stated midpoint and final result are correct. Diagram labels A $(-2,5)$, B $z_1$, C $(4,1)$, D $z_2$, O, Re and Im are retained.

Printed answer key

$z_2=3+6i$.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 4; solution pages 19. These are original document page numbers, retained as text metadata.