Three by One
There are many different methods to solve this geometrical problem - how many can you find?
Age
16 to 18
Challenge level
$ABCD$ is a rectangle where $BC$ = $3AB$. $P$ and $Q$ are points on $BC$ such that $BP$ = $PQ$ = $QC$.
Image
Show that: angle $DBC$ + angle $DPC$ = angle $DQC$
Generalise this result.
N.B. This problem can be tackled in at least 8 different ways using different mathematics learnt in the last two years in school and earlier. The methods are essentially the same when viewed from a more advanced perspective.
Here is a hint to help on you on the way :
Let $\alpha + \beta = \gamma$ and $\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$
where $\alpha = \tan^{-1}{1\over3}$, $\beta = \tan^{-1}{1\over2}$, $\gamma = \tan^{-1}1$.
The solutions produced here by school students show eight different methods. Reflection on these methods will help other students to see something of the 'bigger picture' in a way they will not experience from ploughing through the syllabus and working from textbooks (although that is also absolutely necessary).
Eight distinct proofs were given to this problem by two students, Alex and Neil (Madras College) using respectively sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry.
Alex and Neil went on to generalise this
problem to rectangles with dimensions $n$ by 1.
Why do this problem ?
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 right way.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?
Possible approach
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.
Key questions
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?