An enlargement changes the size of a shape from a fixed centre; a translation slides it without turning. One is described by a scale factor and a centre, the other by a single column vector.
Enlargement: every distance from the centre is multiplied by the scale factor \(k\); describe it with the scale factor and the centre. Translation: every point moves by the same column vector \(\begin{pmatrix}x\\y\end{pmatrix}\) (\(x\) across, \(y\) up); describe it with that vector.
Fig. 1: triangle A is enlarged by scale factor 2 from centre (1, 1); each vertex of B is twice as far from the centre, along the same ray
Section 01
Bigger, smaller, same shape.
In the lab, place the centre of enlargement and set the scale factor to 2, then to \(\frac{1}{2}\). Follow the rays from the centre through each vertex: the image vertex is always on the same ray, \(k\) times as far away. Move the centre and watch the image jump, even though its size stays the same.
Scale factor \(k > 1\) makes the shape bigger; \(0 < k < 1\) makes it smaller. Both are still called enlargements.
Multiply the step from the centre: if a vertex is 3 right and 1 up from the centre, with \(k = 2\) its image is 6 right and 2 up from the centre.
Negative scale factor (Extended): the image is on the OTHER side of the centre and upside down.
Angles stay the same; every length is multiplied by \(k\) (the image is similar, not congruent).
Worked example Scale factor 2 from a centre
Triangle \(A\) has vertices \((2, 2)\), \((4, 2)\), \((2, 3)\). Enlarge it by scale factor 2, centre \((1, 1)\).
Steps from the centre: \((2, 2)\) is 1 right, 1 up; \((4, 2)\) is 3 right, 1 up; \((2, 3)\) is 1 right, 2 up.
Answer: \((2, 1)\), \((4, 1)\), \((4, 3)\): half the size
Worked example A negative scale factor (Extended)
Find the image of \((2, 3)\) after an enlargement, scale factor \(-2\), centre \((1, 1)\).
Step from the centre to the point: 1 right, 2 up.
Multiply by \(-2\): 2 LEFT, 4 DOWN.
From \((1, 1)\): \((1 - 2, 1 - 4) = (-1, -3)\).
Answer: \((-1, -3)\)
Examiner trap
Multiplying the coordinates when the centre is NOT the origin. Enlarging \((2, 2)\) by scale factor 2 from \((1, 1)\) gives \((3, 3)\), not \((4, 4)\). Always measure from the centre.
Section 02
Describing an enlargement.
To describe an enlargement fully you need the scale factor and the centre. The scale factor compares corresponding lengths; the centre is where lines through corresponding vertices meet.
Scale factor \(= \dfrac{\text{image length}}{\text{object length}}\), measured on matching sides.
Centre: draw lines through two or three pairs of corresponding vertices and extend them back; they cross at the centre.
Worked example Finding the scale factor and centre
Triangle \(P\) has vertices \((1, 1)\), \((2, 1)\), \((2, 3)\). Triangle \(Q\) has vertices \((3, 1)\), \((6, 1)\), \((6, 7)\). Describe fully the single transformation that maps \(P\) onto \(Q\).
The line through \((1, 1)\) and \((3, 1)\) is \(y = 1\).
The line through \((2, 3)\) and \((6, 7)\) has gradient 1, so it is \(y = x + 1\); it meets \(y = 1\) at \((0, 1)\).
Check: \((2, 1)\) is 2 right of \((0, 1)\); tripled is 6 right, giving \((6, 1)\). Correct.
Answer: Enlargement, scale factor 3, centre \((0, 1)\)
Worked example Going the other way
Using the triangles above, describe fully the single transformation that maps \(Q\) onto \(P\).
Lengths shrink from 3 to 1, so the scale factor is \(\dfrac{1}{3}\).
The centre is the same point, \((0, 1)\).
Answer: Enlargement, scale factor \(\dfrac{1}{3}\), centre \((0, 1)\)
Examiner trap
Writing the scale factor as 3 (or \(-3\)) when the shape gets smaller. Divide IMAGE length by OBJECT length: going from 3 to 1 gives \(\dfrac{1}{3}\). A negative scale factor means the image is on the other side of the centre, not that it is smaller.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
The point \((3, 2)\) is enlarged by scale factor 2, centre \((0, 0)\). Where is its image?
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A shape is enlarged by scale factor \(\dfrac{1}{2}\). What happens to its side lengths?
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The point \((4, -1)\) is translated by \(\begin{pmatrix}-2\\5\end{pmatrix}\). Where is its image?
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What must you state to describe an enlargement fully?
Section 03
Translation by a vector.
In the lab, change the top and bottom numbers of the column vector and watch the image slide. The top number moves the shape across (positive right), the bottom number moves it up (positive up). The shape never turns or changes size.
Column vector \(\begin{pmatrix}x\\y\end{pmatrix}\): \(x\) across (negative is left), \(y\) up (negative is down).
Every point of the shape moves by the same vector, so the image is congruent and faces the same way.
Describe a translation as 'Translation by the vector \(\begin{pmatrix}x\\y\end{pmatrix}\)'. Never write it as coordinates.
Worked example Translating a point
Translate the point \((2, -1)\) by the vector \(\begin{pmatrix}-3\\4\end{pmatrix}\).
Across: \(2 + (-3) = -1\).
Up: \(-1 + 4 = 3\).
Answer: \((-1, 3)\)
Worked example Finding the vector
A translation maps \((1, 2)\) onto \((5, -1)\). Describe the translation fully.
Across: from 1 to 5 is \(+4\).
Up: from 2 to \(-1\) is \(-3\) (3 down).
Answer: Translation by the vector \(\begin{pmatrix}4\\-3\end{pmatrix}\)
Worked example One after another (Extended)
The point \((3, 1)\) is translated by \(\begin{pmatrix}2\\-4\end{pmatrix}\), and then its image is reflected in the \(x\)-axis. Find the final point.
Translate: \((3 + 2, 1 - 4) = (5, -3)\).
Reflect in the \(x\)-axis: the \(y\)-coordinate changes sign, giving \((5, 3)\).
Answer: \((5, 3)\)
Your turn
A translation maps \((-2, 5)\) onto \((3, 1)\). Write down the column vector.
Across: \(3 - (-2) = 5\). Up: \(1 - 5 = -4\). The vector is \(\begin{pmatrix}5\\-4\end{pmatrix}\).
Examiner trap
Subtracting the wrong way round. The vector from object to image is image minus object: from \((1, 2)\) to \((5, -1)\) is \(\begin{pmatrix}4\\-3\end{pmatrix}\), and \(\begin{pmatrix}-4\\3\end{pmatrix}\) would slide the shape back again.
Stage 2 · Exam
Exam-style questions
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A vertex at \((5, 3)\) is enlarged by scale factor 3, centre \((2, 1)\). Where is its image?
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Triangle \(A\) is enlarged to give triangle \(B\). A side of length 8 cm in \(A\) corresponds to a side of length 2 cm in \(B\). What is the scale factor of the enlargement?
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A translation maps \((3, -2)\) onto \((-1, 4)\). What is the column vector of the translation?
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(Extended) The point \((2, 3)\) is enlarged by scale factor \(-2\), centre \((1, 1)\). Where is its image?
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