Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Here we use excerpts from several different
We start with the solution
fromMolly and Catherine of Mount School, York
Points C, A and A' are co-linear. To explain this, consider the
common tangent to both circles at the point of contact. It is both
perpendicular to CA' and to AA' so, as there is a common point in
A' , the initial statement is demonstrated.
Molly and Catherine's method proves, in the
same way, that C, B and B' are co-linear and A, D and B are
co-linear. They finished the proof correctly as did Sarah and
Caroline of Ipswich High School and James of Hethersett School,
Norfolk. For the following method you may visualise swinging AD
round to AA' and BD round to BB'.
Next is the solution byCharlotte
and Frances of Ipswich High School
It can be proved that the triangle ABC has perimeter equal to
the diameter of the circle centre C because AD = AA' making CAD =
CAA' and BD = BB'
so CBD = CBB' . This proves that the perimeter of the triangle
ADBCA equals the diameter of the circle because CAA' and CBB' are
Clare of Maidstone Girls' Grammar School used
much the same method as Charlotte and Frances.