Similar triangles – measuring height with shadows

What it shows

Two triangles ABC and A'B'C' are kept similar while students reshape, enlarge and rotate them: corresponding angles stay equal and every pair of corresponding sides has the same ratio k. The perimeter ratio is also k, while the area ratio is k². The application mode measures height indirectly. At one moment the sun's rays make the same angle with the ground everywhere, so a person and a tree form two similar right triangles with their shadows (AA similarity), and the tree's height follows from the ratio of the two shadow lengths.

How to use

In Similar triangles mode, drag A, B and C to reshape the triangle and drag B' or C' to change the Ratio of similarity k, turning on Angle measures to compare angles. Check the side, perimeter and area ratios after each change. Then press Measure height with shadows, set the Person's height, and drag the treetop or the tip of its shadow; students calculate the tree's height from the shadows before reading the result.

Parameters you can change

• Ratio of similarity k 0.5–3
• Angle A of the triangle (application mode: angle of the sun's rays) 25–75 °
• Mode Similar triangles, Measuring tree height with shadows
• Person's height (application mode) 1.2–2 m
• Show angle measures

Questions to explore

1. Why are the triangle formed by the person and their shadow and the triangle formed by the tree and its shadow similar?
2. What relationship do the two triangles have when k = 1?
3. If the ratio of similarity is 3, how many times larger is the area of the second triangle?