Transformations and axes of symmetry
When you flip a photo on your phone, slide a piece across a board game or spin a hurley on the pitch, the shape stays the same but its position changes. How can we say exactly where it ends up? In this page you will learn four transformations, and how to spot when a shape looks the same after one.
Translations
A translation slides every point of a shape the same distance in the same direction. The shape does not turn or flip, and it stays the same size.
We describe a translation by a vector $(a, b)$. The number $a$ says how far to move across (right is positive, left is negative). The number $b$ says how far to move up or down (up is positive, down is negative).
The original point is the object and the new point is the image. We write the image of $A$ as $A'$.
$$(x, y) \mapsto (x + a, \; y + b)$$To find the vector from an object to its image, subtract: vector $= (x' - x, \; y' - y)$. To find the object from an image, subtract the vector from the image.
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.
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.
The point $A(5, 2)$ is mapped to $A'(1, 6)$ by a translation. a) Find the translation vector. b) Use it to find the image of $B(-1, 0)$. c) Find the point whose image is…
The point $A(5, 2)$ is mapped to $A'(1, 6)$ by a translation.
- a) Find the translation vector.
- b) Use it to find the image of $B(-1, 0)$.
- c) Find the point whose image is $(7, 7)$.
Central and axial symmetry
Axial symmetry (a reflection) flips a shape in a line, called the axis. Each point and its image are the same distance from the axis, on opposite sides, and the line joining them is perpendicular to the axis.
Central symmetry turns every point through $180^\circ$ about a fixed point, the centre. The centre is the midpoint of each point and its image.
For a point $(x, y)$:
| Transformation | Image |
|---|---|
| Axial symmetry in the $x$-axis | $(x, -y)$ |
| Axial symmetry in the $y$-axis | $(-x, y)$ |
| Axial symmetry in the line $y = x$ | $(y, x)$ |
| Central symmetry in the origin | $(-x, -y)$ |
Why? In the $x$-axis, the height above the axis becomes the same depth below it, so only $y$ changes sign. In the origin, you go the same distance through the centre to the opposite side, so both signs change.
When you apply two transformations one after the other, the order can matter. For example, translate $(1, 0)$ by $(1, 0)$ and then reflect in the $y$-axis: you get $(-2, 0)$. Reflect first and then translate: you get $(0, 0)$. The results are different.
But reflecting in the $x$-axis and the $y$-axis gives the same result in either order: one changes the sign of $y$, the other changes the sign of $x$, and the two changes do not interfere. Try $(3, 1)$ both ways: $(3, 1) \to (3, -1) \to (-3, -1)$ and $(3, 1) \to (-3, 1) \to (-3, -1)$.
Find the image of each point $P(4, -2)$ under: a) central symmetry in the origin b) axial symmetry in the $x$-axis c) axial symmetry in the $y$-axis
Find the image of each point $P(4, -2)$ under:
- a) central symmetry in the origin
- b) axial symmetry in the $x$-axis
- c) axial symmetry in the $y$-axis
Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$ and $C(4, 3)$. a) Find the images of the vertices under central symmetry in the point $(0, 0)$. b) Find the images under…
Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$ and $C(4, 3)$.
- a) Find the images of the vertices under central symmetry in the point $(0, 0)$.
- b) Find the images under axial symmetry in the line $x = 0$.
- c) Find the images under translation by $(-3, 2)$.
The point $A(2, 5)$ is reflected in the $x$-axis and the image is then reflected in the $y$-axis. a) Find the final image $A''$. b) Name the single transformation that…
The point $A(2, 5)$ is reflected in the $x$-axis and the image is then reflected in the $y$-axis.
- a) Find the final image $A''$.
- b) Name the single transformation that maps $A$ directly to $A''$.
- c) Does the order of the two reflections matter? Justify your answer.
Rotations
A rotation (or turn) moves every point through a given angle about a fixed centre. You need the centre, the angle and the direction: clockwise or anticlockwise.
About the origin, anticlockwise:
| Angle | Image of $(x, y)$ |
|---|---|
| $90^\circ$ | $(-y, x)$ |
| $180^\circ$ | $(-x, -y)$ |
| $270^\circ$ | $(y, -x)$ |
A rotation of $180^\circ$ about a point is the same as central symmetry in that point.
The point $P(3, 1)$ is rotated by $90^\circ$ anticlockwise about the origin to give $P'$. a) Find the coordinates of $P'$. b) Find the image of $Q(-2, 4)$ under the same…
The point $P(3, 1)$ is rotated by $90^\circ$ anticlockwise about the origin to give $P'$.
- a) Find the coordinates of $P'$.
- b) Find the image of $Q(-2, 4)$ under the same rotation.
- c) Find the image of $P$ under a rotation of $180^\circ$ about the origin and name another transformation that gives the same image.
Axes of symmetry
A shape has an axis of symmetry if a mirror line along it makes one half fall exactly onto the other. A shape has central symmetry if a $180^\circ$ turn about some point maps it onto itself.
| Shape | Axes of symmetry | Central symmetry? |
|---|---|---|
| Equilateral triangle | 3 | No |
| Isosceles triangle | 1 | No |
| Rectangle (not a square) | 2 | Yes |
| Square | 4 | Yes |
| Rhombus (not a square) | 2 | Yes |
| Parallelogram (not a rectangle or rhombus) | 0 | Yes |
| Regular hexagon | 6 | Yes |
For a regular polygon with $n$ sides there are $n$ axes.
State the number of axes of symmetry of each shape. a) an equilateral triangle b) a rectangle that is not a square c) a square d) a regular hexagon
State the number of axes of symmetry of each shape.
- a) an equilateral triangle
- b) a rectangle that is not a square
- c) a square
- d) a regular hexagon
Describe the symmetry of the following. Give the number of axes of symmetry and say whether each has central symmetry. a) a parallelogram that is not a rectangle or…
Describe the symmetry of the following. Give the number of axes of symmetry and say whether each has central symmetry.
- a) a parallelogram that is not a rectangle or rhombus
- b) a rhombus that is not a square
- c) the capital letters H, T and S
Transformations of shapes and the line $y = x$
To transform a shape, transform each vertex and join the images. Distances and angles do not change under any of the transformations on this page, so the image is the same size and shape as the object.
For the line $y = x$, swap the co-ordinates: $(x, y) \mapsto (y, x)$.
Vuelve al principio: a flipped photo, a slid piece and a turned hurley are a reflection, a translation and a rotation, and each one is described exactly by a rule on the co-ordinates.
Triangle $PQR$ has $P(1, 2)$, $Q(5, 2)$ and $R(3, 6)$. It is reflected in the line $y = x$ to give triangle $P'Q'R'$. a) Find the coordinates of $P'$, $Q'$ and $R'$. b)…
Triangle $PQR$ has $P(1, 2)$, $Q(5, 2)$ and $R(3, 6)$. It is reflected in the line $y = x$ to give triangle $P'Q'R'$.
- a) Find the coordinates of $P'$, $Q'$ and $R'$.
- b) Show that $PQ = P'Q'$ and state which property of axial symmetry this illustrates.
- c) Is $PQR$ symmetrical? State the number of axes of symmetry of triangle $PQR$ and give the equation of one.