Length of a Shadow If a tree 20 ft tall casts a shadow 8 ft long, how long would the shadow of a 30-ft tree be at the same time and place?

The concept of similar triangles states that if two triangles have the same shape, their corresponding angles are equal, and the lengths of their corresponding sides are proportional. In this problem, the heights of the trees and the lengths of their shadows form two similar triangles, allowing us to set up a proportion to find the unknown shadow length.

Proportional relationships occur when two quantities maintain a constant ratio. In this scenario, the ratio of the height of the tree to the length of its shadow remains constant. By establishing this ratio from the first tree, we can apply it to the second tree to determine the length of its shadow.

Ratios express the relationship between two quantities, often written as a fraction. In this question, we can express the height of the trees and the lengths of their shadows as ratios. Understanding how to manipulate and solve these ratios is essential for finding the length of the shadow of the second tree based on the known measurements of the first.