https://file.aiquickdraw.com/mathbot/file/1728280873830580esu8m.jpg

The problem involves using similar triangles to determine the width of a river. Here’s a breakdown of how to solve it:

1. **Identify the Similar Triangles**: The problem shows two right triangles sharing a common angle, so they are similar by AA (Angle-Angle) similarity.

2. **Set Up the Ratios**: For similar triangles, the corresponding sides are proportional. Let \( w \) be the width of the river (the length of the base of the smaller triangle). We have:
   \[
   \frac{w}{31} = \frac{19.9}{58}
   \]

3. **Cross-Multiply and Solve for \( w \)**:
   \[
   w \times 58 = 19.9 \times 31
   \]
   \[
   w = \frac{19.9 \times 31}{58}
   \]

4. **Calculate**:
   - \( 19.9 \times 31 = 616.9 \)
   - Dividing by 58 gives:
     \[
     w \approx 10.64
     \]

5. **Round to the Nearest Meter**: \( w \approx 11 \) meters.

So, the width of the river is approximately **11 meters**.

Would you like a more detailed explanation or have any questions?

Here are some related questions to deepen your understanding:

1. What are the properties of similar triangles?
2. How can you determine if two triangles are similar using AA, SSS, or SAS?
3. How does cross-multiplying help solve proportions?
4. Why is it important to use similar triangles in real-world problems like this?
5. How would the solution change if the height of the smaller triangle was different?

**Tip:** In similar triangles, corresponding angles are equal, and the sides are proportional, which is a powerful tool for solving indirect measurement problems.

Latoya wants to measure the width of a river. She marks off two right triangles, as shown in the figure. The base of the larger triangle has a length of 58 m, and the base of the smaller triangle has a length of 31 m. The height of the smaller triangle is 19.9 m. How wide is the river?

Solve for the width of a river using two similar right triangles. Given a larger triangle with a base of 58 m and a smaller triangle with a base of 31 m and height of 19.9 m, apply geometry and proportions to find the width. Suitable for Grades 8-10.

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