Which principle relates heights and shadows to compute a tree's height in the problem about a tall person and a tree?

• A. Pythagorean theorem
• B. Conservation of energy
• C. Similar triangles
• D. Law of reflection

Answer: (C)

Similar triangles formed by the sun’s rays are the key idea here. When the sun shines, the height of a object and the length of its shadow create a right triangle with the same angle at the sun for any object under the same light. That means the two triangles—the one for the tall person and the one for the tree—are similar, so their height-to-shadow-length ratios are equal.

Use that proportional relationship to find the tree’s height: height of the person divided by the length of the person’s shadow equals the height of the tree divided by the length of the tree’s shadow. If you know the person’s height and both shadow lengths, you can solve for the tree’s height. For example, if the person is 6 feet tall and their shadow is 4 feet, and the tree’s shadow is 12 feet, then the tree’s height is (12/4) × 6 = 18 feet.

Pythagorean theorem isn’t needed here because we’re not calculating a single triangle’s hypotenuse; conservation of energy and law of reflection don’t apply to this geometric measurement.