Why do this
problem?
This is a quick, simple problem on gradients
with a neat result. It will help to reinforce ideas about lines
and coordinate geometry and factorising expressions.
Possible
approach
This problem is ideally used as a lesson starter. It might be
useful when revisiting ideas about coordinate geometry in a
more advanced context. The second part might challenge some
students and could be left as an optional extra for those who
find the first part straightforward.
In the problem, the result is described as 'beautiful'. Do
students see it as such? Can they understand why a
mathematician might see it as beautiful?
Key question
Can you find the gradients of the segments?
Possible extension
Can you find a similar result if the parabola were replaced by
the cubic equation $y=x^3$.
How far can you repeat the analysis if two lines joining two
pairs of points on the parabola were perpendicular?
Possible support
Just do the first part of the problem.