In this one problem you meet many important aspects of mathematics.
It illustrates how much mathematics is inter-related. It shows the
value of not being content to find one solution, but of asking
yourself "could I solve this another way?" or "have I found the
best method?" It is very satisfying to feel you have somehow got to
the very essence of a mathematical idea by looking at it in the
Also this problem provides a good example of how you can generalise
from a result that is really a simple case of a much more general
result. Are you content just to solve a problem or do you ask
yourself "what if ...." and try to find more general results?
You could challenge your class to find as many different methods of
solving this problem asposible. You could tell them that one pair
of school students found 8 different methods. At some stage of this
work you might mention that these two students used respectively
sines, cosines, tangents, vectors, matrices, coordinate geometry,
complex numbers and pure geometry.
Perhaps your students could work in pairs. They could come to the
board and present their solutions to the rest of the class and/or
make posters for the classroom wall. You could see collectively how
many different methods your class can find.
What lengths in the diagram can we find?
What do we know about sines, cosines, tangents, vectors, matrices,
coordinate geometry, complex numbers and pure geometry that we can
use to prove this result?
How might we generalise this result?
The linked article
Why Stop at Three by One?
beautifully generalises this