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To solve this problem, we can apply trigonometry, specifically the Law of Cosines. The triangle \(MOP\) is the key triangle, and we need to find the distance across the lake, which is represented by \(MN\).

### Given:
- \(OM = 115\) m (distance from \(O\) to \(M\))
- \(OP = 104.55\) m (distance from \(O\) to \(P\))
- \(LO = 140\) m (distance from \(O\) to \(L\))
- Angle \( \angle LOM = 90^\circ \) (right angle at \(M\))

### Steps:

1. **Apply the Pythagorean Theorem in the right triangle \(LOM\)** to find \(LM\).

\[
LM^2 = LO^2 - OM^2
\]

\[
LM^2 = 140^2 - 115^2 = 19600 - 13225 = 7375
\]

\[
LM = \sqrt{7375} = 85.89 \, \text{m}
\]

2. **Use the Law of Cosines in triangle \(MNP\)** to find \(MN\).

We need to calculate the distance across the lake, \(MN\), by knowing the triangle \(MNP\).

I'll calculate this step by step.The length of \(LM\) is approximately \(79.84\) meters.

Now, we can proceed by using the Law of Cosines in triangle \(MNP\) to calculate the distance across the lake, \(MN\). We know that \(OM = 115 \, \text{m}\) and \(OP = 104.55 \, \text{m}\), and we need to find \(MN\). To apply the Law of Cosines, we also need the angle between \(OM\) and \(OP\), which we don't have directly.

Let's analyze how to extract that angle or use another approach for distance across the lake. Would you like to proceed with Law of Cosines assuming the angle? Or ask for a different approach?

To indirectly measure the distance across a lake, Bilquis makes use of a couple landmarks at points M and N. She measures LO, OM, and OP as marked. Find the distance across the lake (MN), rounding your answer to the nearest hundredth of a meter.

Learn how to find the distance across a lake using trigonometry and the Law of Cosines. This problem involves measuring the sides and angles of triangles formed around the lake. Detailed solution suitable for students in Grades 10-12.

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