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To solve for the distance \( d \) across the river, we can use the concept of similar triangles. Since the triangles are similar, the corresponding sides are proportional.

The problem states that:
\[
\frac{d}{120} = \frac{300}{50}
\]

### Step 1: Set up the proportion
The proportion based on the similarity of triangles is:
\[
\frac{d}{120} = \frac{300}{50}
\]

### Step 2: Simplify the right side
Calculate \(\frac{300}{50}\):
\[
\frac{300}{50} = 6
\]

So the proportion becomes:
\[
\frac{d}{120} = 6
\]

### Step 3: Solve for \( d \)
Multiply both sides by 120 to isolate \( d \):
\[
d = 6 \times 120
\]
\[
d = 720
\]

### Final Answer
The distance across the river is \( \mathbf{720 \, \text{feet}} \).

Would you like more details, or do you have any questions?

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Here are five related questions to deepen your understanding:

1. How would the answer change if the length on the other side were different?
2. Can you explain why the triangles are similar in this context?
3. What would happen if the triangles were not similar?
4. How could you verify the solution using another method?
5. How does proportional reasoning help in solving problems involving similar triangles?

**Tip:** Always check if triangles are similar before applying proportions, as this is only valid for similar triangles.

Find the distance, d, across the river. Assume that the triangles are similar and the ratio of d to 120 ft is the same as the ratio of 300 ft to 50 ft.

Learn how to find the distance across a river by applying proportions in similar triangles. This problem uses the ratio of d to 120 ft, equated to the ratio of 300 ft to 50 ft, to determine the unknown distance. Ideal for students in Grades 7-9, this problem reinforces the concept of similar triangles and proportional reasoning.

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