https://file.aiquickdraw.com/mathbot/file/1727370156250kpms25nq.jpg

The problem involves using the AA Triangle Similarity Theorem to calculate the width of the river, where two triangles formed in the diagram are similar. Let's break down the given information to find the width of the river:

### Key points and measurements:
- The width of the river is the distance from Point A to Point B, denoted as \( AB \), which we are tasked to find.
- \( AC = 8 \, \text{ft} \) is the distance walked parallel to the river.
- \( DE = 20 \, \text{ft} \), and this is the distance walked parallel to the river after turning 90°.
- \( BD = 10 \, \text{ft} \), which forms part of the triangle.
- \( AE = 25 \, \text{ft} \).

### Using triangle similarity:
The AA (Angle-Angle) Triangle Similarity Theorem tells us that the two triangles, \( \triangle ABD \) and \( \triangle ADE \), are similar by AA similarity (since they share angles).

Given the similar triangles, the ratio of corresponding sides is equal. Therefore:

\[
\frac{AB}{BD} = \frac{AE}{DE}
\]

Substitute the known values:

\[
\frac{AB}{10} = \frac{25}{20}
\]

Simplifying:

\[
\frac{AB}{10} = \frac{5}{4}
\]

Now solve for \( AB \):

\[
AB = \frac{5}{4} \times 10 = 12.5 \, \text{ft}
\]

Thus, the width of the river is **12.5 feet**.

---

Would you like more details or further explanation? Here are some related questions:

1. How does the AA similarity theorem apply to this situation?
2. Why is it necessary for the triangles to be similar in this method?
3. Can this method be used in real-life situations to measure other distances indirectly?
4. What would change in the calculation if the angles were different?
5. How could this setup be altered to use another triangle similarity theorem like SAS?

**Tip:** When solving problems with similar triangles, always double-check that you correctly identified the corresponding sides.

An anthropologist studying an ancient society reaches a river that needs to be crossed. She locates a tree directly across the river at Point A, marks her location as Point B, and then turns and walks to Point C. She continues walking until Points A and C are directly in line with her final location, Point E. What is the width of the river?

Learn how to calculate the width of a river using the AA Triangle Similarity Theorem. This problem involves using proportions between similar triangles to find the unknown distance. The solution provides detailed steps and is suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation