The lines given by the following four equations enclose a square.

  1. \(y-2=x\)

  2. \(y+x=6\)

  3. \(y=x-1\)

  4. \(y+x-3=0\)

You might like to convince yourself of this before going any further!

  • Given any three of the four lines that enclose a square, can you find the other one?
  • You are given the area of the square and the coordinates of one vertex. Can you find possible equations of the four lines enclosing it?
  • What is a minimal amount of information needed to be able to describe the equations of the four lines enclosing a square?

You might like to use the following scenarios as starting points for your investigation.

Three of the four equations are

  1. \(y=3x+2\)

  2. \(3y+x=8\)

  3. \(3y+x=12\)

Find the fourth equation.

Find the area of the square.

The coordinates of two adjacent vertices of a square are \(\left(\frac{1}{5},\frac{3}{5}\right)\) and \(\left(\frac{1}{5},\frac{-3}{5}\right)\).

Find four equations that could enclose the square.

Find the area of the square.

One vertex of a square is located at \(\left(1,\frac{1}{5}\right)\) and one side adjacent to this vertex lies on the line \(5y=x\).

Can you describe a corresponding set of three equations that would enclose a square with an area of exactly \(1\)?

You might like to try a graphical approach to this problem, perhaps using Desmos or GeoGebra.