Formula sheet
Scale factor
k = PA'/PA is given on the exam formula sheet. You must still measure both distances from the same centre P, put the image distance on top, and know that k has no unit.
What the symbols mean
- k Scale factor, how many times bigger the image is than the object (no unit).
- PA' Distance from the centre of enlargement P to an image point A' (length unit).
- PA Distance from the centre of enlargement P to the matching object point A (length unit).
Given in the exam, or memorise?
k = PA'/PA is given on the exam formula sheet. You must still measure both distances from the same centre P, put the image distance on top, and know that k has no unit.
Why it works
Enlargement stretches every distance from the centre by the same factor.
- Under an enlargement, a point A moves to A' along the line from centre P.
- The distance PA' is a fixed multiple of PA for every point.
- That fixed multiple is the scale factor, so k = PA'/PA.
Worked example 1
Under an enlargement with centre P, an object point A is 3 cm from P and its image A' is 12 cm from P. Find the scale factor.
- k = PA'/PA
- = 12 ÷ 3
- = 4
Worked example 2
An enlargement has centre P and scale factor 3. An object point A is 5 cm from P.
How far is its image A' from P?
- k = PA'/PA, so PA' = k × PA
- PA' = 3 × 5
- = 15
Where students go wrong
- Inverting the ratio: writing PA/PA' instead of PA'/PA (image over object).
- Measuring the two distances from different points instead of the same centre P.
- Attaching a unit to k, the scale factor is a pure number.
- Forgetting a negative scale factor when the image is on the opposite side of P.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Which distance goes on top, image or object?
The image distance PA' goes on top and the object distance PA on the bottom: k = PA'/PA. An easy memory aid is that the primed letter (the image) sits above.
Flip them and you get the reciprocal, which describes shrinking back to the object rather than the enlargement itself.
Does the scale factor have a unit?
No. Since k divides one length by another length, the units cancel and k is a pure number.
A scale factor of 4 means every distance from the centre becomes four times as long. Writing '4 cm' for k is a mistake, it is simply 4, telling you how many times bigger the image is.
What does a scale factor bigger than 1 or between 0 and 1 mean?
If k is greater than 1 the image is larger than the object, a true enlargement. If k is between 0 and 1 the image is smaller, a reduction.
If k is negative the image also flips to the opposite side of the centre. The size of k tells you the stretch, and its sign tells you the side.