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Cambridge IGCSE Mathematics — 0580 Extended

Topic 7.4: Transformations — Enlargement

An enlargement changes size, keeping the shape. You need a centre and a scale factor \(k\). Rays from the centre through each vertex of the object also pass through the matching image vertex, and the image is \( |k| \) times as far from the centre as the object is.

Positive scale factor

If \(k > 1\) the image is larger and further from the centre, on the same side. From the origin, \((x, y) \to (kx, ky)\). From a general centre \(C\), multiply the vector \(\overrightarrow{C\,P}\) by \(k\).

Method

  1. Mark the centre. Draw a ray from the centre through each vertex.
  2. Measure the distance centre → vertex. Multiply by \(k\) and mark that distance along the same ray.
  3. Join the image vertices. Sides of the image are parallel to the object.

Enlarge triangle \(ABC\) by scale factor 2 from the origin.

Triangle enlarged by scale factor 2 from the origin along rays through each vertex
\(OA' = 2 \times OA\). Image sides are parallel to the object and twice as long.

Enlarge triangle \(ABC\) by scale factor 2 from centre \((-2, 1)\).

Triangle enlarged by scale factor 2 from centre (minus 2, 1)
Vector centre\(\to A\) is \((3, 0)\); twice that is \((6, 0)\), so \(A' = (4, 1)\).

Fractional scale factor

If \(0 < k < 1\) the image is a reduction: smaller, and closer to the centre, still on the same side of the centre.

Enlarge triangle \(ABC\) by scale factor \(\tfrac{1}{2}\) from the origin.

Triangle reduced by scale factor one half from the origin
\(A(4, 2) \to A'(2, 1)\). The image is similar; lengths are multiplied by \(\tfrac{1}{2}\).

Negative scale factor

A negative \(k\) puts the image on the opposite side of the centre. Distances are still multiplied by \(|k|\), and the shape comes out inverted through the centre. From \(O\), \(k = -2\) sends \((x, y)\) to \((-2x, -2y)\).

Enlarge triangle \(ABC\) by scale factor \(-2\) from the origin.

Triangle enlarged by scale factor minus 2, landing on the opposite side of the origin
\(A(2, 1) \to A'(-4, -2)\). A negative \(k\) turns the shape through the centre.
Two enlargements of the same triangle from the origin, k = 2 and k = minus 2
Same \(|k|\) and same centre — negative \(k\) sends the image through the centre.

Scale factor \(-1\) about a centre is the same image as a \(180^\circ\) rotation about that centre. On 0580, if the question is in the enlargement section you still call it an enlargement, scale factor \(-1\).

Describing an enlargement

A full description needs the centre and the scale factor (with sign). Join each object vertex to its image; the rays meet at the centre. Then

\[ k = \dfrac{\text{image length}}{\text{object length}} \]

Give \(k\) a minus if object and image are on opposite sides of the centre.

Find the centre and scale factor that maps \(ABC\) onto \(A'B'C'\).

Rays AA-prime BB-prime CC-prime concurrent at the centre (1, 2)
Scale factor = image length ÷ object length (minus if they are on opposite sides).

Paper 2 (non-calculator)

Count squares along a ray: if the object is 2 squares from the centre and the image is 4 squares on the same ray, \(k = 2\). If the image is 4 squares the other way, \(k = -2\).

Try this

Centre \(O\). Point \(A(3, -1)\) is enlarged by scale factor \(-2\). Write \(A'\). What scale factor maps \(A'\) back to \(A\)?

Show answer
Answer

  1. Multiply both coordinates by \(-2\)

    \[ A' = (-6, 2) \]

  2. The inverse enlargement uses \(1/k\)

    \(k = -\tfrac{1}{2}\), same centre \(O\)

Exam Traps

  • An enlargement is not fully described by the scale factor alone. The centre is required.
  • A negative scale factor is not a rotation in the mark scheme unless the question asks for a single equivalent transformation and rotation is the intended name.

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