Similar triangles measure heights and distances that cannot be reached directly.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Shadow problems
At one moment the sun makes the same angle everywhere, so an object and its shadow form triangles similar to any other object and its shadow.
The mirror method
A mirror on the ground reflects at equal angles, creating two similar triangles between the viewer and the object.
Scale drawings
A map or plan is a dilation of reality. A scale of 1 to 500 means every drawn length stands for 500 real ones.
Careful with areas on maps
At a scale of 1 to 500, areas are in the ratio 1 to 250000. The square catches people out constantly.
Watch the units
Both parts of a ratio must be in the same unit. Mixing centimetres with metres is the standard error here.
Measuring what you cannot reach
Shadow lengths, mirrors on the ground and sighting devices all create similar triangles. The height of a tree follows from a proportion, which is one of the oldest practical uses of geometry.
Scale drawings and models
Maps, blueprints and models are similar figures. The scale is the ratio, and converting between drawing and reality is applying it in one direction or the other.
Areas square and volumes cube
A model at one-tenth scale has one-hundredth the surface area and one-thousandth the volume. This is why scale models cannot simply be scaled up into working structures.
Check the answer is plausible
A tree computed as 3 metres or 300 metres from a plausible shadow indicates an inverted ratio. Sanity-checking against the situation catches proportion errors that the algebra will not.
Step 2: Try It Yourself
Tap and try it out.
Before: 3
After: 12
The factor is more than 1, so the bar got longer.
Step 3: Watch an Example
One step at a time.
Watch Diego Measure a Tree
A 2 m post casts a 3 m shadow. A tree casts a 21 m shadow at the same moment.
- Step 1
The sun is at the same angle for both, so the two triangles are similar by AA.
Step 4: Your Turn
Practice makes it stick.
The Tree
Problem 1 of 2
A 2 m post casts a 3 m shadow. A tree casts a 21 m shadow. How tall is the tree, in metres?
m
The Map
Problem 2 of 2
A map has scale 1 to 500. A drawn length of 4 cm stands for how many centimetres?
Measure Indirectly
1 of 8
A 3 m post casts a 4 m shadow. A tree casts a 20 m shadow. Tree height in metres?
2 of 8
A 2 m post casts a 5 m shadow. A building casts a 40 m shadow. Height in metres?
3 of 8
Scale 1 to 200. A drawn 5 cm stands for how many centimetres?
4 of 8
Scale 1 to 50. A real 600 cm is drawn as how many centimetres?
5 of 8
At scale 1 to 100, areas are in what ratio? Enter the second number.
6 of 8
A 1 m stick casts a 2 m shadow. A pole casts a 14 m shadow. Height in metres?
7 of 8
Put the shadow method in order.
- 1Conclude the two triangles are similar by AA.
- 2Set the height to shadow ratios equal.
- 3Cross-multiply and solve for the unknown height.
- 4Note that the sun is at the same angle for both objects.
8 of 8
Scale 1 to 25. A real 300 cm is drawn as how many centimetres?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A 2 m post casts a 3 m shadow. A tower casts a 30 m shadow. Height in metres?
Question 2 of 2
At a scale of 1 to 500, what is the ratio of areas?
What You Learned
- Similar triangles measure heights that cannot be reached.
- Shadows at the same moment create similar triangles by AA.
- On a scale drawing, areas scale by the square of the length ratio.