Step by step
The question
Question: find the height x of the tree.
A 2 m pole has a 3 m shadow.
At the same time, the tree has a 12 m shadow.
1 Two similar triangles
The sun's rays come in at the same angle.
So the two dashed lines are parallel.
The pole and the tree both stand straight up: a right angle at the bottom.
Same angles, so the two triangles are similar.
One is a bigger copy of the other.
2 Match the sides
Matching sides go together.
The height goes with the height.
The shadow goes with the shadow.
3 Find the scale factor
The scale factor k says how many times bigger.
We know both shadows: 12 m and 3 m.
k is big over small.
Careful!
Never put a height with a shadow.
Shadow with shadow, height with height.
12 divided by 3 is 4.
The tree's triangle is 4 times bigger.
4 Find x
x is in the big triangle.
Its partner is the pole's height, 2 m.
So x is 2 times 4.
That is 8.
So the height of the tree is 8 m.
5 Check
Let's check: x with its partner.
8/2 is 4, the scale factor.
The pole is shorter than its shadow.
So the tree is shorter than its shadow too.
It makes sense!
Let's look back.
First: two similar triangles.
Then: match the sides.
Then: find the scale factor.
Then: find x.
Last: check.