I am a student in 10th grade and we were discussing total internal reflection in class today. My teacher told us that when the angle of incidence is equal to the critical angle total internal reflection will occur. However, if the refracted ray is at 90 should it not travel precisely on the line where both mediums meet? If that is true how do we decide which medium the refracted ray is in and will the properties of the wave change?
The way the question is phrased it does not relate to a real world situation and so any answer is hypothetical unless it relates to the real world.
For example in the real world, a single ray with no width cannot be produced, the interface cannot be completely flat, the angle of refraction cannot be exactly $90^\circ,\,\ldots $
If you try the experiment you will find sometimes that some of the beam passes through the interface and there is also partial reflection and at other times there is only a reflected beam.
I think that the best one can do is to state what happens when the angle of incidence is just less than the critical angle and when the angle of incidence is just greater than the critical angle.
As a 10th grader you might consider the following question.
How does a light ray incident at an angle greater than the critical angle "know" that it has reached an interface without actually venturing forth into the interface?
The answer is that there is an evanescent_wave on the other (forbidden) side of the interface and this wave extends a few wavelengths into the interface.
The Wikipedia article gives an example of a practical demonstartion which is very difficult to do because the evanescent waves penetrate such a small distance through an interface.
However using microwaves which have wavelength of the order of centimetres the existance of evanescent waves can be shown relatively easily and you might have access to such equipment in your laboratory?
It can happen only to those rays that are at the angular threshold of being refracted and totally reflected. In general, rays are the tangential envelopes to the lines that orthogonal to the equal phase surfaces of the propagating waves. The situation you are describing can only happen with an ideal, that is, infinite extent plane wave, but that is incompatible with a two media interface unless the rays themselves, all of them, are perpendicular to the interface plane between the two media.