Another way to measure the height of a tree uses its shadow....

Another way to measure the height of a tree uses its shadow. The measurer walks away from the tree along its shadow until her head is in line with the top of the tree and the tip of the shadow. (FIGURE CAN'T COPY) Suppose that $C D=6$ ft and $B D=20$ ft. What other distance would you need to know in order to determine the tree's height?

Another way to measure the height of a tree uses its shadow. The measurer walks away from the tree along its shadow until her head is in line with the top of the tree and the tip of the shadow. (FIGURE CAN'T COPY)
Suppose that $C D=6$ ft and $B D=20$ ft. What other distance would you need to know in order to determine the tree's height?

Key Concepts

Indirect Measurement Using Shadows

This method measures the height of an object by comparing it to a smaller, measurable object (in this case, the observer) using their shadows. The principle relies on the sun’s rays being parallel, leading to the formation of similar triangles, and it shows how measurements can be taken indirectly when direct measurement is impractical.

Distance Between the Tree and the Observer

To complete the proportional relationship, the total shadow of the tree must be known, which is the sum of the observer’s shadow length (given as 20 ft) and the distance between the tree’s base and the observer’s feet. Therefore, knowing this additional ground distance is crucial for determining the tree’s full shadow length, and in turn, its height.

Similar Triangles

In this problem, the observer, the tree’s top, and the tip of the tree’s shadow form one triangle, while the observer’s height and shadow create a smaller, similar triangle. Recognizing similar triangles is essential because it allows the use of proportional relationships between corresponding sides to solve for unknown lengths.

Proportional Relationships

Once similar triangles have been identified, corresponding side ratios are equal. This means that the ratio of the tree’s height to its shadow is equal to the ratio of the observer’s height to her shadow. This proportion is the key to setting up an equation to solve for the unknown tree height, provided all but one distance (the unknown) is known.

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