Step by step
The question
Question: two similar shapes have matching lengths 2 and 5, and the small area is 8: find the large area.
lengths 2 and 5, small area = 8
1 Length scale factor
The scale factor k: every length of the large one is 5/2 times the small one.
k = 52
2 Area factor
Area grows in two directions: k times k.
That is 25/4.
area factor = k² = (52)² = 254
Careful!
The area does not grow like the lengths.
It grows by k squared.
Careful: not 8 × 5/2 ✗
3 Use it
8 times 25/4 is 50.
large area = 8 × 254 = 50
4 Check
Let's check: the large area divided by the small one is 25/4, which is k squared.
So it is right!
50 ÷ 8 = 25/4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
large area = 8 × 254 = 50