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In this problem, we are tasked with finding the distance \( PR \), which is across the river.

### Given Information:
- \( OE = 275 \, \text{ft} \)
- \( RE = 210 \, \text{ft} \)
- \( OP = 115 \, \text{ft} \)
- The angle \( \angle POR \) is right-angled, so \( \triangle POR \) is a right triangle.

Since \( \triangle POR \) is a right triangle, we can use the Pythagorean Theorem to calculate the distance \( PR \), which is the hypotenuse of the triangle.

### Pythagorean Theorem:
\[
PR^2 = OP^2 + OR^2
\]

We are given \( OP = 115 \, \text{ft} \), but we do not have \( OR \) directly. However, we can use \( \triangle ORE \), which is a right triangle, to find \( OR \) using the Pythagorean Theorem again.

### Step 1: Find \( OR \) using \( \triangle ORE \)
\[
OR^2 + RE^2 = OE^2
\]
\[
OR^2 + (210)^2 = (275)^2
\]
\[
OR^2 + 44100 = 75625
\]
\[
OR^2 = 75625 - 44100 = 31525
\]
\[
OR = \sqrt{31525} \approx 177.58 \, \text{ft}
\]

### Step 2: Find \( PR \) using \( \triangle POR \)
Now that we have \( OR \), we can find \( PR \).
\[
PR^2 = OP^2 + OR^2
\]
\[
PR^2 = (115)^2 + (177.58)^2
\]
\[
PR^2 = 13225 + 31525
\]
\[
PR^2 = 44750
\]
\[
PR = \sqrt{44750} \approx 211.57 \, \text{ft}
\]

### Final Answer:
The distance across the river, \( PR \), is approximately **212 feet** when rounded to the nearest foot.

Would you like more details on any step? 

Here are 5 related questions:
1. How can the Law of Sines or Cosines be used to solve a similar problem?
2. What other methods can be used to measure distances across rivers or obstacles?
3. How would errors in the measurements affect the final result?
4. How would you apply trigonometry to find angles in this scenario?
5. Can you solve for another side of the triangle if more angles were given?

**Tip**: In right-angle triangle problems, always check if you can apply the Pythagorean theorem before other trigonometric methods.

To indirectly measure the distance across a river, Sebastian stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Sebastian draws the diagram below to show the lengths and angles that he measured. Find PR, the distance across the river. Round your answer to the nearest foot.

Learn how to find the distance across a river using trigonometry and the Pythagorean Theorem. In this problem, Sebastian measures lengths and angles, then applies the Pythagorean Theorem to calculate the hypotenuse of a right triangle. Suitable for students in Grades 9-12.

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