Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
Hint 1 : Remember, we need nice simple ratios between any two
strings in your scale.
For example 5/9 of the unit would probably sound good with Note
1 and 6/7 would also sound OK with Note 1, but 5/9 and 6/7 would
probably not sound good together because their length ratio (35:54)
is not simple.
Hint 2 : Once you have the ratios for all your strings - you
should be able to show that taking any pair of strings, you only
once have a ratio with numbers bigger than 10.
Hint 3: You have ratios based on halves and thirds - what else
could you try?