Exercises: Transformations and axes of symmetry

The point $A(2, 3)$ is translated by the vector $(4, -1)$. Find the image $A'$. Then translate $B(-3, 5)$ by the same vector.

Find the image of each point $P(4, -2)$ under:

1. a) central symmetry in the origin
2. b) axial symmetry in the $x$-axis
3. c) axial symmetry in the $y$-axis

State the number of axes of symmetry of each shape.

1. a) an equilateral triangle
2. b) a rectangle that is not a square
3. c) a square
4. d) a regular hexagon

Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$ and $C(4, 3)$.

1. a) Find the images of the vertices under central symmetry in the point $(0, 0)$.
2. b) Find the images under axial symmetry in the line $x = 0$.
3. c) Find the images under translation by $(-3, 2)$.

The point $P(3, 1)$ is rotated by $90^\circ$ anticlockwise about the origin to give $P'$.

1. a) Find the coordinates of $P'$.
2. b) Find the image of $Q(-2, 4)$ under the same rotation.
3. c) Find the image of $P$ under a rotation of $180^\circ$ about the origin and name another transformation that gives the same image.

The point $A(5, 2)$ is mapped to $A'(1, 6)$ by a translation.

1. a) Find the translation vector.
2. b) Use it to find the image of $B(-1, 0)$.
3. c) Find the point whose image is $(7, 7)$.

Triangle $PQR$ has $P(1, 2)$, $Q(5, 2)$ and $R(3, 6)$. It is reflected in the line $y = x$ to give triangle $P'Q'R'$.

1. a) Find the coordinates of $P'$, $Q'$ and $R'$.
2. b) Show that $PQ = P'Q'$ and state which property of axial symmetry this illustrates.
3. c) Is $PQR$ symmetrical? State the number of axes of symmetry of triangle $PQR$ and give the equation of one.

The point $A(2, 5)$ is reflected in the $x$-axis and the image is then reflected in the $y$-axis.

1. a) Find the final image $A''$.
2. b) Name the single transformation that maps $A$ directly to $A''$.
3. c) Does the order of the two reflections matter? Justify your answer.

Describe the symmetry of the following. Give the number of axes of symmetry and say whether each has central symmetry.

1. a) a parallelogram that is not a rectangle or rhombus
2. b) a rhombus that is not a square
3. c) the capital letters H, T and S