A transformation moves or resizes a shape. This note covers the four transformations in the course (translation, reflection, rotation and enlargement) and shows how to describe each one fully in words, including how to find a centre of rotation or enlargement. Describing a transformation completely is a skill that needs a precise checklist.
Key ideas
The original shape is the object and the new shape is the image. A point's image is usually labelled with a dash, so \(A\) maps to \(A'\). Reflections, rotations and translations keep the shape the same size and shape (the object and image are congruent). Enlargements keep the shape but change the size (the object and image are similar).
To describe a transformation fully, you must give the name and the details that fix it:
- Translation: state the column vector \(\begin{pmatrix}x\\y\end{pmatrix}\). Every point moves by the same vector, and the shape does not turn.
- Reflection: state the equation of the mirror line, such as \(x=3\), \(y=-1\), \(y=x\) or \(y=-x\). Each point and its image are the same distance from the mirror line, on opposite sides, and the line joining them is perpendicular to the mirror line.
- Rotation: state the angle, the direction (clockwise or anticlockwise) and the centre of rotation. Common angles are \(90^\circ\), \(180^\circ\) and \(270^\circ\). A \(180^\circ\) turn needs no direction because both ways give the same result.
- Enlargement: state the scale factor and the centre of enlargement. The distance of each image point from the centre is the scale factor times the distance of the object point from the centre.
For an enlargement with scale factor \(k\):
- If \(k\gt1\) the image is larger. If \(0\lt k\lt1\) (a fraction), the image is smaller, even though it is still called an enlargement.
- If \(k\) is negative (Higher tier), the image is on the opposite side of the centre and is turned upside down. For example, scale factor \(-2\) makes the image twice as big and rotated \(180^\circ\) about the centre.
- Lengths are multiplied by \(|k|\). Areas are multiplied by \(k^2\).
Finding the centre. For an enlargement, draw a straight line through each object point and its image point. The lines meet at the centre. For a rotation, draw the perpendicular bisector of the line joining a point to its image, and do the same for a second point. The two bisectors cross at the centre of rotation. Check the angle by comparing the lines from the centre to a point and to its image.
A useful way to decide which transformation you are looking at: if the image is a different size, it is an enlargement. If the shape is the same size and the same way up, it is a translation. If it is the same size but turned, it is a rotation. If it is the same size and flipped (a mirror image), it is a reflection. Always check at least two points.
You can also describe a point's image by a rule. A reflection in the \(x\)-axis changes \((x,y)\) to \((x,-y)\). A reflection in the \(y\)-axis changes it to \((-x,y)\). Reflection in \(y=x\) swaps the coordinates to \((y,x)\). A rotation of \(90^\circ\) anticlockwise about the origin changes \((x,y)\) to \((-y,x)\), and a rotation of \(180^\circ\) about the origin gives \((-x,-y)\).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Object, image | The starting shape and the shape after the transformation |
| Translation by \(\begin{pmatrix}a\\b\end{pmatrix}\) | \((x,y)\to(x+a,\,y+b)\) |
| Reflection in line \(m\) | Mirror line \(m\) is the perpendicular bisector of each point and its image |
| Rotation | Needs angle, direction, centre |
| Enlargement, scale factor \(k\), centre \(C\) | \(CP'=k\times CP\) (measured as vectors from \(C\)) |
| Scale factor \(k\) | Lengths \(\times |k|\), areas \(\times k^2\) |
| Congruent / similar | Same size and shape / same shape, different size |
Worked examples
Example 1 Triangle \(T\) has vertices \((1,1)\), \((4,1)\) and \((4,3)\). Its image has vertices \((3,1)\), \((0,1)\) and \((0,3)\) respectively. Describe fully the single transformation.
The image is the same size, so it is not an enlargement. In \(T\) the right angle is at \((4,1)\), with the base going left and the vertical side going up. In the image the right angle is at \((0,1)\), with the base going right and the vertical side still going up. A rotation would not keep the sides facing like this, so the shape has been flipped. This is a reflection.
Midpoint of \((1,1)\) and \((3,1)\) is \((2,1)\). Midpoint of \((4,1)\) and \((0,1)\) is \((2,1)\). Midpoint of \((4,3)\) and \((0,3)\) is \((2,3)\). All have \(x=2\), and each joining line is horizontal, so perpendicular to the vertical line \(x=2\).
Answer: reflection in the line \(x=2\).
Example 2 Triangle \(A\) has vertices \((2,1)\), \((4,1)\) and \((4,3)\). Triangle \(B\) has vertices \((1,-2)\), \((1,-4)\) and \((3,-4)\) respectively. Describe fully the single transformation that maps \(A\) to \(B\).
The size is unchanged and the shape is turned, so it is a rotation.
The perpendicular bisector of \((2,1)\) and \((1,-2)\): the midpoint is \((1.5,-0.5)\) and the joining line has direction \((-1,-3)\), so the bisector is \(x+3y=0\).
The perpendicular bisector of \((4,1)\) and \((1,-4)\): the midpoint is \((2.5,-1.5)\) and the direction is \((-3,-5)\), so the bisector is \(3x+5y=0\).
Both lines pass through \((0,0)\), and the only point on both is the origin. The centre is \((0,0)\).
The line from the origin to \((2,1)\) has gradient \(\tfrac12\), and the line to its image \((1,-2)\) has gradient \(-2\). Since \(\tfrac12\times(-2)=-1\), the lines are perpendicular, so the angle is \(90^\circ\). The point moves from the first quadrant to the fourth, which is clockwise.
Answer: rotation of \(90^\circ\) clockwise about \((0,0)\).
Example 3 Triangle \(P\) has vertices \((2,3)\), \((5,3)\) and \((5,7)\). Its image \(P'\) has vertices \((3,5)\), \((9,5)\) and \((9,13)\). Describe fully the transformation.
The image is larger, so it is an enlargement.
Scale factor: the base of \(P\) has length \(5-2=3\) and the base of \(P'\) has length \(9-3=6\). Scale factor \(=6\div3=2\).
Centre: the line through \((2,3)\) and \((3,5)\) is \(y=2x-1\). The line through \((5,3)\) and \((9,5)\) is \(y=\tfrac12x+\tfrac12\). Setting \(2x-1=\tfrac12x+\tfrac12\) gives \(\tfrac32x=\tfrac32\), so \(x=1\) and \(y=1\).
Check: from \((1,1)\), the vector to \((2,3)\) is \((1,2)\) and the vector to \((3,5)\) is \((2,4)=2\times(1,2)\). It works.
Answer: enlargement, scale factor 2, centre \((1,1)\).
Common errors
- Describing a transformation with only its name, such as "rotation", with no angle, direction or centre. A full description needs all the details.
- Giving two transformations in one answer. A question that asks for a single transformation wants only one.
- Giving the mirror line as "the \(y\)-axis" when it should be an equation, or writing \(y=2\) when the line is vertical (\(x=2\)).
- Measuring the enlargement from the wrong point. Distances are measured from the centre of enlargement, not from a corner of the shape.
- Using "enlargement" for a scale factor between 0 and 1 but then writing a scale factor larger than 1. A smaller image means a fractional scale factor.
Exam tip: Test your answer with two vertices before you write it down. Use tracing paper for a rotation, and for an enlargement draw the lines through the vertices to check that they meet at the centre you have named.
Check yourself
- Reflect the point \((3,-2)\) in the line \(y=x\).
Answer
Reflection in \(y=x\) swaps the coordinates, so the image is \((-2,3)\). - Rotate the point \((4,1)\) through \(90^\circ\) anticlockwise about the origin.
Answer
The rule is \((x,y)\to(-y,x)\), so \((4,1)\to(-1,4)\). - Describe the translation that maps \((2,5)\) to \((-1,9)\).
Answer
Subtract the coordinates: \(-1-2=-3\) and \(9-5=4\). The translation is by the vector \(\begin{pmatrix}-3\\4\end{pmatrix}\). - A triangle has a vertex at \((3,4)\). It is enlarged by scale factor 3 with centre \((1,2)\). Find the image of that vertex.
Answer
The vector from the centre to the vertex is \((3-1,\,4-2)=(2,2)\). Multiply by 3 to get \((6,6)\). Add this to the centre: \((1+6,\,2+6)=(7,8)\).