Given information:
- The person is swimming at an angle of 135 degrees to the current of the river.
- The person wants to reach the opposite point.
- We need to find the ratio of the person's velocity to the velocity of the river water.

Finding the Solution:

Step 1: Draw the diagram of the situation
- Draw a river flowing from left to right.
- Mark the starting point of the person on the left bank of the river.
- Mark the opposite point on the right bank of the river.
- Draw the velocity vector of the river water in the direction of the river flow.
- Draw the velocity vector of the person in the direction of 135 degrees to the river current.

Step 2: Resolve the velocities into components
- Resolve the person's velocity vector into two components: one parallel to the river current and one perpendicular to the river current.
- The component parallel to the river current will not contribute to the person's progress towards the opposite point, so we can ignore it.
- The component perpendicular to the river current will contribute to the person's progress towards the opposite point.

Step 3: Apply trigonometry
- Use trigonometry to find the component of the person's velocity perpendicular to the river current.
- Let the magnitude of the person's velocity be Vp and the magnitude of the river water velocity be Vr.
- The angle between the person's velocity vector and the river current is 45 degrees (since the person is swimming at 135 degrees to the river current).
- The component of the person's velocity perpendicular to the river current is Vp sin 45 degrees = Vp / √2.
- The component of the river water velocity perpendicular to the person's velocity is Vr sin 45 degrees = Vr / √2.

Step 4: Use the relative velocity concept
- Use the concept of relative velocity to find the net velocity of the person towards the opposite point.
- The net velocity of the person towards the opposite point is the difference between the component of the person's velocity perpendicular to the river current and the component of the river water velocity perpendicular to the person's velocity.
- Net velocity = (Vp / √2) - (Vr / √2) = (Vp - Vr) / √2.

Step 5: Use the given information to find the ratio of velocities
- We know that the person wants to reach the opposite point, so the net velocity towards the opposite point must be equal to the speed of the person's progress towards the opposite point.
- Let the distance between the starting point and the opposite point be D.
- The time taken by the person to reach the opposite point is D / (Vp - Vr) / √2.
- We can now use the given information to find the ratio of velocities:
- Time taken by the person to reach the opposite point = Distance / Speed.
- Speed = Distance / Time taken by the person to reach the opposite point.
- Speed = D / (Vp - Vr) / √2.
- Speed of the person's progress towards the opposite point = Vp / √2.
- Ratio of velocities = (Vp / √2) / (D / (Vp - Vr) / √2) = Vp / (D / (Vp - Vr)).
- Ratio of velocities = Vp