Why do this problem?
The problem naturally brings in the formula for a circle and it
could be used to introduce this formula.
The problem requires knowledge and understanding of the formula
for the distance between two points, the use of Pythagoras
Theorem and algebraic skills in solving simultaneous equations.
The problem introduces the concept of orthogonality which is very
important in higher mathematics.
Possible approach
Depending on their experience of problem solving and thinking for
themselves, if there is some lack of confidence then the class
could first discuss how they are going to set about solving the
problem. This should be in the context of the teacher making it
explicit that the learners should be developing problem solving
skills and growing out of dependence on the teacher to indicate
methods to use.
- List what information is given, what they are tryng to
find, and then how to use the information given.
- In trying to find the orthogonal circle the 3 unknowns are
its radius and the coordinates of its centre.
- The diagram suggests the use of 3 right angled triangles
and the use of Pythagoras' Theorem to give 3 equations.
- How can they solve the equations. How can they eliminate
one of the variables between pairs of equations.
The learners can then work independently having planned how to
work on the problem.
Key questions
What information is given in the question?
What are the unknowns? Have we assigned notation to refer to
the unknowns?
Can we use the given information, and our mathematical
knowledge, to write down some equations?
What methods have we used in the past to solve similar
equations? Are those methods useful here?