Use similar triangles to solve. A person who is 6 feet tall is standing 154 feet from the base of a tree, and the tree casts a 165 foot shadow. The person's shadow is 11 feet in length. What is the height of the tree? ft CHE 154 ft 6 ft 11 f

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
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**Using Similar Triangles to Solve for the Height of a Tree**

**Problem Statement:**
A person who is 6 feet tall is standing 154 feet from the base of a tree. The tree casts a 165-foot shadow. The person's shadow is 11 feet in length. What is the height of the tree?

**Diagram Explanation:**
- The diagram shows two similar triangles.
- The smaller triangle consists of:
  - A vertical line representing the person who is 6 feet tall.
  - A horizontal line representing the person’s shadow, which is 11 feet long.
- The larger triangle consists of:
  - A vertical line representing the tree.
  - A horizontal line representing the tree's shadow, which is 165 feet long, starting from a distance of 154 feet from the tree's base.

**Solution:**
- Use the properties of similar triangles to set up a proportion:
  \[
  \text{Height of the Person} / \text{Length of Person's Shadow} = \text{Height of the Tree} / \text{Length of Tree's Shadow}
  \]
- Substitute the known values:
  \[
  6 \text{ ft} / 11 \text{ ft} = \text{Height of the Tree} / 165 \text{ ft}
  \]
- Solve for the height of the tree.

**Conclusion:**
By solving the proportion, you can determine the height of the tree using the similarity of triangles.
Transcribed Image Text:**Using Similar Triangles to Solve for the Height of a Tree** **Problem Statement:** A person who is 6 feet tall is standing 154 feet from the base of a tree. The tree casts a 165-foot shadow. The person's shadow is 11 feet in length. What is the height of the tree? **Diagram Explanation:** - The diagram shows two similar triangles. - The smaller triangle consists of: - A vertical line representing the person who is 6 feet tall. - A horizontal line representing the person’s shadow, which is 11 feet long. - The larger triangle consists of: - A vertical line representing the tree. - A horizontal line representing the tree's shadow, which is 165 feet long, starting from a distance of 154 feet from the tree's base. **Solution:** - Use the properties of similar triangles to set up a proportion: \[ \text{Height of the Person} / \text{Length of Person's Shadow} = \text{Height of the Tree} / \text{Length of Tree's Shadow} \] - Substitute the known values: \[ 6 \text{ ft} / 11 \text{ ft} = \text{Height of the Tree} / 165 \text{ ft} \] - Solve for the height of the tree. **Conclusion:** By solving the proportion, you can determine the height of the tree using the similarity of triangles.
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