Step by step
The question
Question: two similar shapes have matching lengths 2 and 5, and the small area is 12: find the large area.
1 Length scale factor
Similar means the same shape: one is a bigger copy.
The scale factor k says how many times longer: the large one over the small one, 5/2.
2 Area factor
Area grows in two directions: k times k.
On top, 5 times 5 is 25.
At the bottom, 2 times 2 is 4.
That is 25/4.
Why?
For example, a square 1 by 1, made 2 times longer both ways, holds 4 little squares, not 2.
3 Use it
Going to the large one, we multiply: 12 times 25/4.
Times 25/4 means times 25, then divide by 4.
We can do them in any order.
Keep the numbers small: 12 divided by 4 is 3.
Then 3 times 25 is 75.
So the large area is 75.
Careful!
The area does not grow like the lengths.
It grows by k squared.
4 Check
Let's check: the large area divided by the small one is 25/4, which is k squared.
So it is right!
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.