Congruency, Enlargement and Combined Transformations · Form 5
What a scale factor does to length vs area
Under an enlargement with scale factor k, every length is multiplied by k, but the area is multiplied by k², because area is two-dimensional (length × length).
Length grows by k, area by k²
When a shape is enlarged by scale factor k, each side length becomes k times as long. Area, however, is length × length, so it is multiplied by k × k = k².
For example, k = 3 makes every side 3 times longer but the area 9 times larger, not 3 times.
Reading it backwards, from area to length
Questions often give the ratio of areas and ask for the scale factor of the lengths, or the other way round. Because the area factor is k², you take the square root to recover the length factor: an image with 4 times the area came from k = √4 = 2.
The key skill is recognising which 'factor' a number refers to, length or area, before you use it.
The usual misconception
The most common error is multiplying the area by k instead of k². Enlarging by scale factor 2 does not double the area; it makes it 2² = 4 times larger.
Note too that a scale factor between 0 and 1 (say ½) is a reduction, lengths shrink and area shrinks to k² = ¼ of the original, while the area factor k² stays positive even when k is negative.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
If the scale factor is k, how does the area change?
Every length in the image is k times the corresponding length in the object, but area scales by k² because area involves two dimensions multiplied together. For example, scale factor 3 makes lengths 3 times bigger but area 9 times bigger.
How do I find the image's area if I only know the object's area and scale factor?
Multiply the object's area by k², not k, to get the image's area: area of image = k² × area of object. Students who multiply by k only get the area wrong - this is one of the most common SPM errors on this topic.
If the scale factor is negative or a fraction, does area still use k²?
Yes - the area scale factor is always k², or |k|² for negative k, giving a positive value since area cannot be negative. A fractional k, such as k = 1/2, still gives an area scale factor of 1/4, meaning the image is smaller than the object.