An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
Madras College, St. Andrews and Natalie from West Flegg Middle
School, Norfolk correctly used
congruent (n.b. not similar) triangles to show the sameness, having
first constructed an altitude through C parallel to TS and PU. When
equal angles and side lengths (SA and AC) are identified it is soon
apparent that triangle CVA is congruent to triangle ATS and also
that triangle CBV is congruent to BPU. Hence ST + PU =
Jack and Jan
at Necton Middle School "discovered that triangle STA together
with triangle BUP, when rotated, will fit exactly into triangle
ACB........." but don't state about which
centres the rotations are to take place.
Another interesting approach here came
without a name on it, from West Flegg MS again. This time after
drawing CV perpendicular to TU, if you draw two squares one with
side TV and the other with side VU , boxing in the original two
squares, it is clear to see that ST + PU = AV + VB = AB.