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### Number and algebra

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### Advanced mathematics

# Enclosing Squares

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### Parallel Lines

### Reflecting Lines

Links to the University of Cambridge website
Links to the NRICH website Home page

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Age 11 to 14

Challenge Level

Here's a problem to work at with your graphic calculator or graph-plotting package on a computer.

If you plot the following lines

$\begin{eqnarray} y &=& 2x + 1\\ y &=& 2x + 4 \\ y &=& -0.5x + 1\\ y &=& -0.5x + 2.5 \end{eqnarray}$ |

the lines will enclose a square.

Can you find other sets of sloping lines that enclose a square?

If you are given the equations of two parallel lines

$y = ax + b$ and $y = ax + c$

can you explain how to find the equations of the other two
lines that would enclose a square, if you know that one of the
vertices is at $(0,b)$?

[In the example at the top, $a = 2$, $b = 1$ and $c =
4$]

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.