The points \(A(1,1)\), \(B(5,1)\), \(C(5,3)\) and \(D(1,3)\) are the vertices of a rectangle. (a) Write down the equations of the two lines of symmetry of t...

(a) The two lines of symmetry. A rectangle has exactly two mirror lines, each passing through the midpoints of a pair of opposite sides. Find each one by averaging the coordinates of the sides it lies between.

• The vertical sides are at \(x=1\) (through \(A\) and \(D\)) and \(x=5\) (through \(B\) and \(C\)), so the vertical mirror line is \( x = \frac{1+5}{2} = 3 \), that is \( x=3 \) [B1].
• The horizontal sides are at \(y=1\) (through \(A\) and \(B\)) and \(y=3\) (through \(D\) and \(C\)), so the horizontal mirror line is \( y = \frac{1+3}{2} = 2 \), that is \( y=2 \) [B1].

Equivalent forms of these equations are accepted. The diagonals are not lines of symmetry, because the rectangle is 4 units long and 2 units high, so reflecting in a diagonal would map a side of length 4 onto a side of length 2.

(b) Order of rotational symmetry. The two mirror lines cross at \((3,2)\), the centre of the rectangle. A half turn about that centre sends \(A\) to \(C\) and \(B\) to \(D\), leaving the rectangle unchanged, and the full turn also works. A quarter turn fails because the side lengths differ. The order is 2 [B1].