Age 11 to 14 Challenge Level:
Lorna from Sutton High School sent us her
work on this problem. Well done, Lorna!
When a triangle is rolled along a horizontal line (the baseline),
the paths of the vertices can be described as below.
Equilateral triangle ABC - The overall pattern of paths forms three
overlapping arcs, the radii of which are the triangle side's
length. The arcs all rest upon the same horizontal line, and each
arc starts a triangle side's length away from the one before. The
paths intersect after the vertices concerned have both travelled
Right angled isosceles triangle ABC - The overall pattern of paths
this time consists of two different sized arcs - small arcs (radius
of equal sides AB and AC) and big arcs (radius of side BC). The
path of vertex A only ever forms small arcs, the vertex being
formed by the two equal short sides. Vertices B and C form a
pattern of arcs first big, then small, then big etc., because these
vertices are both connected to one long and one short side. No two
paths ever follow the same track.
Right angled scalene triangle ABC - This time the pattern consists
of three different sized arcs, each the radius of a different
triangle side. Each path consists of two arc sizes (either big and
medium, big and small or medium and small) because each vertex is
made from the joining of two sides of different lengths.
Overall Explanation - When a vertex reaches the baseline, it
becomes the pivot that the triangle rotates around. Therefore it
stays on the baseline for one rotation. Because there are three
sides, each vertex first reaches the baseline every third rotation,
but being a pivot means that it does not make a path. This is why
each path is made up of a pattern of only two different