Indirect measurement with similar triangles involves representing an inaccessible length or distance through two triangles known to have the same shape, often because corresponding angles are equal or both are right triangles formed by parallel lines or shadows. The learner identifies corresponding sides, applies a consistent scale factor or proportion with compatible units, and interprets the solution geometrically rather than matching sides by position alone; the treatment is limited to proportional side relationships in concrete and diagram-based configurations, not trigonometric methods or more advanced similarity theory.
Indirect measurement uses a known length to find an inaccessible length. The idea is to form two similar triangles, meaning they have the same shape, so their corresponding side lengths are proportional.
A person who is meters tall casts a -meter shadow. At the same time, a tree casts a -meter shadow. How tall is the tree?
Identify the two triangles.
Each triangle has:
Both are right triangles, and the sun’s rays are parallel. Therefore, the triangles are similar.
Match corresponding sides.
The person’s height corresponds to the tree’s height.
The person’s shadow corresponds to the tree’s shadow.
So use the proportion
Substitute the known measurements.
Let be the tree’s height:
Solve for .
Multiply both sides by :
State the answer with units.
The tree is approximately tall.
The important point is to match sides that play the same geometric role—not simply sides that appear in the same position in a drawing. Here, both heights correspond, and both shadows correspond.
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