Similarity in practice: height from a shadow, an inaccessible distance
Use similar triangles for measuring in the field: the height of a tree from its shadow, a distance across a river, and the ratio of areas of similar figures. Lay solutions out as Given, Find, Solution, Answer.
- Russia: grade 8
Guide
In the Russian programme this is Grade 8 geometry, chapter "Similar triangles": similarity tests, proportional segments and practical applications (measuring on the ground).
The idea: right triangles with an equal acute angle are similar, so their sides are proportional. The sun's rays are parallel, so an object and its shadow make a triangle with the same angles as a stick and its shadow. For an inaccessible distance, draw on paper a triangle with the same angles and find the scale factor.
The areas of similar figures are in the ratio k2. The figures on the page are not to scale.
The "why" question asks for your own explanation; then compare it with the model.