Transformations: reflection, translation and rotation

A transformation moves every point of a shape to a new place. The shape we get is the image of the shape.

• The image of the point $\displaystyle A$ is written $\displaystyle A'$ (read “A prime”).
• Three transformations: reflection, translation and rotation.

A reflection in a line flips the shape like a mirror: every point goes to the other side of the line, at the same distance from it.
In the drawing, triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a reflection in the line $\displaystyle MN$.

• The segment $\displaystyle AA'$ is perpendicular to the line $\displaystyle MN$.
• $\displaystyle A$ and $\displaystyle A'$ are at the same distance from the line $\displaystyle MN$.
• A point on the line itself stays where it is.

When a reflection in a line takes a shape onto itself, the line is a line of symmetry of the shape: fold the shape along it, and the two halves match.

• A square has four lines of symmetry: its two diagonals, and the two lines through the midpoints of opposite sides.
• The letter A has one line of symmetry: the upright line through its middle.
• The letter F has none.

A translation moves every point of the shape the same distance in the same direction.
In the drawing, triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a translation.

• The segments $\displaystyle AA'$, $\displaystyle BB'$ and $\displaystyle CC'$ are equal and parallel.
• In a translation the shape neither flips over nor turns.
• On squared paper a translation is given in squares: here 4 squares to the right and 1 square up.

A rotation about a point $\displaystyle O$ turns the shape the way a clock hand turns about its pin.
In the drawing, triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a rotation about $\displaystyle O$ by $\displaystyle 90^{\circ}$ anticlockwise.

• Every point stays at the same distance from $\displaystyle O$: $\displaystyle {OA}={OA'}$.
• The angle $\displaystyle AOA'$ is the angle of rotation.
• A rotation also has a direction: clockwise or anticlockwise.
• The point $\displaystyle O$, the centre of rotation, stays where it is.

Reflections, translations and rotations change no length and no angle: the shape moves to another place and keeps its size and its shape.

• Imagine the shape drawn on a sheet of tracing paper: you slide the sheet (a translation), turn it about a pin (a rotation) or turn it over (a reflection).
• The sheet neither stretches nor shrinks, so no length and no angle changes.
• In a triangle and its image: $\displaystyle {A'B'}={AB}$, $\displaystyle {B'C'}={BC}$, $\displaystyle {C'A'}={CA}$, and $\displaystyle {\angle A'}={\angle A}$, $\displaystyle {\angle B'}={\angle B}$, $\displaystyle {\angle C'}={\angle C}$.

Triangle $\displaystyle ABC$ in the drawing is moved 3 squares to the left and 4 squares down.
What are the coordinates of $\displaystyle A'$, $\displaystyle B'$ and $\displaystyle C'$?

Triangle $\displaystyle ABC$ from Exercise 1 is reflected.
Write the coordinates of $\displaystyle A'$, $\displaystyle B'$ and $\displaystyle C'$.

A reflection in the $\displaystyle y$-axis.

A reflection in the $\displaystyle x$-axis.

What happens to the coordinates of a point in a reflection in the $\displaystyle y$-axis? And in the $\displaystyle x$-axis?

Triangle $\displaystyle ABC$ from Exercise 1 is rotated about the origin $\displaystyle O(0,0)$ by $\displaystyle 180^{\circ}$.
Draw the triangle and its image on the squared paper, and write the coordinates of $\displaystyle A'$, $\displaystyle B'$ and $\displaystyle C'$.

Hint: in a rotation by $\displaystyle 180^{\circ}$, the points $\displaystyle A$, $\displaystyle O$, $\displaystyle A'$ lie on one line, and $\displaystyle O$ is halfway between $\displaystyle A$ and $\displaystyle A'$.

How many lines of symmetry does the shape have? Describe them.

A rectangle that is not a square.

An equilateral triangle.

A circle.