Worked examples
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.
k = 123
Careful!
Never put a height with a shadow.
Shadow with shadow, height with height.
Careful: height with height, shadow with shadow
12 divided by 3 is 4.
The tree's triangle is 4 times bigger.
k = 4
4 Find x
x is in the big triangle.
Its partner is the pole's height, 2 m.
So x is 2 times 4.
x = 2 × 4
That is 8.
So the height of the tree is 8 m.
x = 8 m
5 Check
Let's check: x with its partner.
8/2 is 4, the scale factor.
82 = 4
The pole is shorter than its shadow.
So the tree is shorter than its shadow too.
It makes sense!
2 < 3, 8 < 12
Let's look back.
First: two similar triangles.
Then: match the sides.
Then: find the scale factor.
Then: find x.
Last: check.
▶ Watch it solved
The question
Question: find the height x of the tree.
A 2 m pole has a 5 m shadow.
At the same time, the tree has a 15 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: 15 m and 5 m.
k is big over small.
k = 155
Careful!
Never put a height with a shadow.
Shadow with shadow, height with height.
Careful: height with height, shadow with shadow
15 divided by 5 is 3.
The tree's triangle is 3 times bigger.
k = 3
4 Find x
x is in the big triangle.
Its partner is the pole's height, 2 m.
So x is 2 times 3.
x = 2 × 3
That is 6.
So the height of the tree is 6 m.
x = 6 m
5 Check
Let's check: x with its partner.
6/2 is 3, the scale factor.
62 = 3
The pole is shorter than its shadow.
So the tree is shorter than its shadow too.
It makes sense!
2 < 5, 6 < 15
Let's look back.
First: two similar triangles.
Then: match the sides.
Then: find the scale factor.
Then: find x.
Last: check.
▶ Watch it solved
The question
Question: find the shadow x of the tree.
A 1.6 m person has a 2 m shadow.
At the same time, the tree is 12 m tall.
1 Two similar triangles
The sun's rays come in at the same angle.
So the two dashed lines are parallel.
The person 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 heights: 12 m and 1.6 m.
k is big over small.
k = 121.6
Careful!
Never put a height with a shadow.
Height with height, shadow with shadow.
Careful: height with height, shadow with shadow
12 divided by 1.6 is 7.5.
The tree's triangle is 7.5 times bigger.
k = 7.5
4 Find x
x is in the big triangle.
Its partner is the person's shadow, 2 m.
So x is 2 times 7.5.
x = 2 × 7.5
That is 15.
So the shadow of the tree is 15 m.
x = 15 m
5 Check
Let's check: x with its partner.
15/2 is 7.5, the scale factor.
152 = 7.5
The person is shorter than their shadow.
So the tree is shorter than its shadow too.
It makes sense!
1.6 < 2, 12 < 15
Let's look back.
First: two similar triangles.
Then: match the sides.
Then: find the scale factor.
Then: find x.
Last: check.
▶ Watch it solved
