I've had a similar problem and also needed the geometries of the output for further calculations. I tryed the above solutions, but what worked best for me in the end was the extraction of line segments using buffer zones.

1. Open "Buffer" from Processing Toolbox and choose a distance of 5000 m:

2. Go to Vector->Processing->Intersection and select your river layer as Input Layer and the created Buffer layer as overlay layer:

3. This will create a layer with the described line segments of 10 km each around the midpoint. This output layer is also useful for further geometry calculations (length, sinuosity,...). I mention this because the above "Service Area (From Layer)"-solution didn't return the correct length in my case.

What worked best for me in the end was the extraction of line segments using buffer zones.

1. Open "Buffer" from Processing Toolbox and choose a distance of 5000 m:

2. Go to Vector->Processing->Intersection and select your river layer as Input Layer and the created Buffer layer as overlay layer:

3. This will create a layer with the described line segments of 10 km each around the midpoint. This output layer is also useful for further geometry calculations (length, sinuosity,...). I mention this because the above "Service Area (From Layer)"-solution didn't return the correct length in my case.