Learn Before
Finding a Tree's Height Using Shadow Lengths and Similar Triangles
Similar figures can be used to determine heights that cannot be measured directly. When the sun casts shadows at the same time of day, a person and a nearby tall object (such as a tree) form two similar triangles — one small and one large — because the sun's rays strike the ground at the same angle.
Problem: Tyler is feet tall. Late one afternoon, his shadow was feet long. At the same time, the shadow of a tree was feet long. Find the height of the tree.
- Read the problem and draw the figure. The small triangle is formed by Tyler (height ft) and his shadow ( ft). The large triangle is formed by the tree (height ) and its shadow ( ft).
- Identify: The height of the tree.
- Name: Let the height of the tree.
- Translate: Because the two triangles are similar, corresponding sides are proportional:
- Solve: Multiply both sides by :
- Check: Tyler's height ( ft) is less than his shadow's length ( ft), so the tree's height should also be less than its shadow's length: ✓. Substituting back: , which simplifies to ✓.
- Answer: The height of the tree is feet.
This example illustrates a practical use of similar triangles: when two objects cast shadows at the same time of day, the ratio of each object's height to its shadow length is constant. Knowing the height and shadow length of the shorter object and the shadow length of the taller object is enough to determine the taller object's height through a single proportion.
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Finding a Tree's Height Using Shadow Lengths and Similar Triangles
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Learn After
A field technician is using the shadow method to estimate the height of a utility pole. According to the principles of similar triangles, what condition must be met for the ratio of the pole's height to its shadow to be equal to the ratio of a reference marker's height to its shadow?
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