Two points plus one line
Draw a line (considered endless in both directions), put a point
somewhere on each side of the line. Label these points A and B. Use
a geometric construction to locate a point, P, on the line, which
is equidistant from A and B.
Age
14 to 16
Challenge level
Problem
Draw a line (considered endless in both directions), put a
point somewhere on each side of the line. Label these points $A$
and $B$.
Use a geometric construction to locate a point, $P$, on the
line, which is equidistant from $A$ and $B$.
Image
Can this point $P$ always be found for any position of $A$ or
$B$ ?
If you believe that this is true how might you construct a
proof?
If it is false, identify the circumstances when no point $P$,
equidistant from $A$ and $B$, exists.
Draw a new arrangement of one line and two points, one on each
side of the line.
Imagine creating a collection of similar arrangements.
Can you suggest useful parameters which would uniquely define
or identify each arrangement?
Parameters are the numbers (measurements or ratios) you might
communicate to another person, say over the telephone, if you
wanted them to produce exactly the same arrangement as your
own.
These parameters define each arrangement.
Mark a point $O$ somewhere on the line and express the length
$OP$ in terms of your parameters.
Getting Started
Simple hint: how would you find the set of points that are
equidistant from two given points?
When might that locus not intersect with the given line?
Harder, last part: try dropping perpendiculars from $A$ and $B$ onto the given line.
The distance between those positions might be useful.
When is $P$ between those positions and when is it outside?
When might that locus not intersect with the given line?
Harder, last part: try dropping perpendiculars from $A$ and $B$ onto the given line.
The distance between those positions might be useful.
When is $P$ between those positions and when is it outside?
Student Solutions
Well done Graham from Feilding High in New
Zealand and also to Andrei from Tudor Vianu National College,
Bucharest, who both gave good answers to the first part of the
problem.
The geometrical solution is to construct the perpendicular bisector of the line $AB$ and mark where it will cross the original line.
This solution works for all cases except where the line $AB$ is itself perpendicular to the original line AND the original line is not the perpendicular bisector of the line $AB$
Can someone now suggest parameters for the general arrangement so that any particular arrangement can be uniquely identified?
The geometrical solution is to construct the perpendicular bisector of the line $AB$ and mark where it will cross the original line.
This solution works for all cases except where the line $AB$ is itself perpendicular to the original line AND the original line is not the perpendicular bisector of the line $AB$
Can someone now suggest parameters for the general arrangement so that any particular arrangement can be uniquely identified?
Teachers' Resources
This problem gives an opportunity to find the parameters that
express the generality in the context.
For example, if the distance between the feet of perpendiculars dropped from $A$ and $B$ is taken as the unit length, then the length of the perpendiculars can be expressed in terms of that unit.
There may be other useful unit lengths to consider, but the important skill is to see that enlarging the diagram doesn't change the feature being explored, so a thoughtful choice of unit can help to make the problem more convenient to handle and access.
For example, if the distance between the feet of perpendiculars dropped from $A$ and $B$ is taken as the unit length, then the length of the perpendiculars can be expressed in terms of that unit.
There may be other useful unit lengths to consider, but the important skill is to see that enlarging the diagram doesn't change the feature being explored, so a thoughtful choice of unit can help to make the problem more convenient to handle and access.