Find a square using a quarter-turn

A, B, C and D represent −2+5i,z1,4+i,z2-2+5i,z_1,4+i,z_2 respectively. ABCD is a square labelled anticlockwise. Show z1=−1z_1=-1 and find z2z_2. marks

[2016/VJC/Prelim/II/3]

Solution

Find z₁ and z₂

Quarter-turn about a vertex

A,B,C,DA,B,C,D represent −2+5i-2+5i, z1z_1, 4+i4+i, z2z_2. ABCDABCD is a square labelled anticlockwise. Show that z1=−1z_1=-1, and find z2z_2.

Explore this idea

Hint

  1. BA→\overrightarrow{BA} is BC→\overrightarrow{BC} turned 90∘90^\circ anticlockwise.

  2. Solve for z1z_1.

  3. Use the diagonals, which bisect each other, to find z2z_2.

Answer · Find z₁ and z₂

z1=−1,z2=3+6iz_1=-1,z_2=3+6i.

Explore the idea

Only a right angle gives a square

Fix AA and CC and slide the angle that BB makes at the centre. Only at 90∘90^\circ do all four sides match and the figure becomes a square.

Angle between the diagonals90°
Try an example
B
-1
AB
5.099
BC
5.099
  • Moves as the slider changes; a faint path shows where it goes
Key idea

A square is a rectangle whose diagonals are also perpendicular. The quarter-turn encodes exactly that.

Ideas to try
  1. At 90∘90^\circ, AB=BCAB=BC: the figure is a square. B=−1B=-1 and D=3+6iD=3+6i.
  2. At 60∘60^\circ the figure is only a rectangle, because AB≠BCAB\ne BC.
Takeaway

Multiply by ii to turn

Right angles in an Argand diagram become multiplication by ±i\pm i. Equal diagonals bisecting each other give a quick check.