Indirect Measurement
Use similar triangles to find lengths that are impossible or impractical to measure directly — heights of buildings, widths of rivers, and distances to unreachable objects.
Introduction
How tall is a flagpole if you can't climb it? How wide is a river if you can't swim across? By setting up similar triangles with measurable sides, you can compute the unknown measurement using a simple proportion. This technique — indirect measurement — has been used for thousands of years.
Past Knowledge
AA Similarity (7.2.1). Proportions & cross-multiplying (7.1.1). Shadow & angle concepts.
Today's Goal
Set up and solve indirect measurement problems using shadows and mirrors.
Future Success
Geometric mean (7.3.2), trigonometry (Unit 8), surveying applications.
Key Concepts
Shadow Method
Two objects standing upright at the same time on flat ground create shadows. The sun hits both at the same angle, so the triangles formed by (object, shadow, sun ray) are similar by AA.
Mirror Method
Place a mirror on the ground between you and the object. The law of reflection creates two right triangles with equal angles of incidence and reflection.
General Setup
- Identify two similar triangles (find two matching angles)
- Label all known measurements
- Set up a proportion with corresponding sides
- Cross-multiply and solve
Worked Examples
Shadow Problem
A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 22-foot shadow. How tall is the tree?
Both triangles share the sun's angle and have 90° at the ground → AA Similarity.
Cross-multiply: →
The tree is 33 feet tall.
Mirror Method
You place a mirror on the ground 30 feet from a building. Standing 5 feet from the mirror, your eye height is 5 feet and you can see the top of the building in the mirror. How tall is the building?
Angle of incidence = angle of reflection → AA Similarity.
feet
The building is 30 feet tall.
River Width
To find the width of a river, you mark points: directly across from a tree , then walk 20 m along the bank to , and 4 m further to . From , you sight the tree and it aligns with a stake at that is 5 m from the bank. Find the river width .
(where includes the river width) by AA.
→
Cross-multiply: → m
The river is 30 meters wide.
Common Pitfalls
Mixing Up Corresponding Sides
Height must pair with height, and shadow must pair with shadow. Don't cross them: is wrong.
Assuming Flat Ground
The shadow and mirror methods assume the ground is level. On a slope, the triangles are not similar because the 90° angle at the ground is broken.
Real-Life Applications
Thales and the Great Pyramid
Around 600 BCE, the Greek mathematician Thales measured the height of the Great Pyramid of Giza by comparing his shadow to the pyramid's shadow — the first recorded use of indirect measurement via similar triangles.
Forestry — Tree Heights
Foresters use clinometers (angle-measuring tools) to sight the top of a tree and create a triangle with known distances and angles. Similar triangles let them estimate tree heights without climbing.
Practice Quiz
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