The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
Prove that the internal angle bisectors of a triangle will never be
perpendicular to each other.
Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST
and PU are perpendicular to AB produced. Show that ST + PU = AB
Consider the ten trapezia. Add an eleventh trapezium to one edge of the arch.
To move from the left of the arch to the right, each tarpezium has to turn the same number of degrees.
Consider the blue edgeof the bottom left hand stone. To complete the arch it has to turn a total of 180 degrees in 10 steps. That is 18 degrees for each step, or 9 degrees for each of the base angles of each trapezium.
Therefore the largest angles of each trapezium are (90+9) degrees and the smallest (90-9)degrees = 81 degrees .
This problem is taken from the UKMT Mathematical Challenges.