Apply triangle similarity to solve practical problems in surveying, architecture, and everyday measurements using the power of proportional relationships.
Using similar triangles to measure tall objects
Principle:
When the sun's rays are parallel, objects and their shadows form similar triangles.
Example:
A 2-meter stick casts a 3-meter shadow. A tree casts a 15-meter shadow. How tall is the tree?
Solution:
• Stick height/Stick shadow = Tree height/Tree shadow
• 2/3 = h/15
• Cross multiply: 2 × 15 = 3 × h
• 30 = 3h → h = 10 meters
Principle:
Place a mirror on the ground and use the reflection to create similar triangles.
Example:
A person 1.8m tall stands 2m from a mirror. They can see the top of a building 20m from the mirror. How tall is the building?
Solution:
• Person height/Person distance = Building height/Building distance
• 1.8/2 = h/20
• Cross multiply: 1.8 × 20 = 2 × h
• 36 = 2h → h = 18 meters
Measuring distances using similar triangles
Scenario:
You need to measure the width of a river without crossing it. You can only measure on your side of the bank.
Method:
1. Mark point A on your side
2. Walk 50m to point B
3. Mark point C across the river
4. Walk 25m to point D
5. Measure angle at D to find similar triangles
Calculation:
If triangles are similar:
River width/50 = 25/50
River width = 25 meters
Scenario:
A map has a scale of 1:25,000. Two cities are 8 cm apart on the map. What is the actual distance?
Scale Factor:
1 cm on map = 25,000 cm in reality
1 cm = 250 meters
1 cm = 0.25 km
Calculation:
Map distance = 8 cm
Actual distance = 8 × 0.25 = 2 km
Answer: 2 kilometers
Creating and using scale models in architecture and engineering
Purpose:
Architects create scale models to visualize buildings before construction and test design concepts.
Example:
A building model is built at 1:100 scale. If the model's height is 15 cm, what is the actual building height?
Solution:
Scale factor = 100
Model height = 15 cm
Actual height = 15 × 100 = 1,500 cm = 15 meters
Purpose:
Engineers use scale models to test structures, bridges, and machines before building full-size versions.
Example:
A bridge model at 1:50 scale has a span of 20 cm. What is the actual bridge span?
Solution:
Scale factor = 50
Model span = 20 cm
Actual span = 20 × 50 = 1,000 cm = 10 meters
Complex problems using multiple similarity concepts
Scenario:
A surveyor needs to measure the height of a cliff. They set up a baseline 100m from the cliff base and measure angles.
Given:
• Baseline distance: 100m
• Angle of elevation: 45°
• Surveyor's eye height: 1.5m
Solution:
Using 45° triangle properties:
Height = Distance = 100m
Total cliff height = 100 + 1.5 = 101.5m
Scenario:
A photographer wants to capture a building that's 30m tall. They need to know how far to stand to fit it in their frame.
Given:
• Building height: 30m
• Camera angle of view: 60°
• Camera height: 1.5m
Solution:
Using trigonometry:
Distance = Height / tan(30°) = 28.5 / 0.577
Distance ≈ 49.4 meters
Apply similarity concepts to solve real-world problems
Given: A 1.8m tall person casts a 2.4m shadow. At the same time, a tree casts a 12m shadow.
Find the height of the tree using similar triangles.
Solution:
• Person height/Person shadow = Tree height/Tree shadow
• 1.8/2.4 = h/12
• Cross multiply: 1.8 × 12 = 2.4 × h
• 21.6 = 2.4h → h = 9 meters
Answer: The tree is 9 meters tall.
Given: A model car is built at 1:24 scale. The model is 15 cm long and 6 cm wide.
Find the actual dimensions of the real car.
Solution:
• Scale factor = 24
• Actual length = 15 × 24 = 360 cm = 3.6 m
• Actual width = 6 × 24 = 144 cm = 1.44 m
Answer: The real car is 3.6m long and 1.44m wide.