Step by step
The question
Question: two similar shapes have matching lengths 3 and 4, and the large area is 48: find the small area.
lengths 3 and 4, large area = 48
1 Length scale factor
The scale factor k: every length of the large one is 4/3 times the small one.
k = 43
2 Area factor
Area grows in two directions: k times k.
That is 16/9.
area factor = k² = (43)² = 169
Careful!
The area does not grow like the lengths.
It grows by k squared.
Careful: not 48 × 4/3 ✗
3 Use it
Going to the small one, we divide: 48 divided by 16/9 is 27.
small area = 48 ÷ 169 = 27
4 Check
Let's check: the large area divided by the small one is 16/9, which is k squared.
So it is right!
48 ÷ 27 = 16/9 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
small area = 48 ÷ 169 = 27