| Solution:
Draw a line segment
horizontally to the left from O, meeting line AP at point Z.
Triangle ZON is isosceles,
because its angles are 67.5, 45, 67.5. So the length of ZO equals
the length of NO, which is 24.
Triangles AZO and APC are
similar, and the length of AC is twice the length of AO, so the length of
PC is twice the length of ZO.
That means the length of PC
is 48.
Alternative solution:
This solution doesn't require you to draw another line, but you need a
little trigonometry...
ADN is isosceles, because its angles are 67.5, 45, 67.5, so ND=AD=BC
OD is (sqrt(2)/2)(BC) because it's half the diagonal of the square, so
NO=(1-sqrt(2)/2)(BC)
From right triangle ABP we see that
BP is the side of the square times the tan of 22.5º.
tan 2x = 2 tan x/(1-tan² x), so
tan 45º = 2 tan 22.5º/(1-tan² 22.5º)
Now let y = tan 22.5º, and of course
tan 45º = 1, so
1 = 2y/(1-y²)
y²+2y-1 = 0
y = -1±sqrt(2)
The positive value of y is the tan of 22.5º (the negative value is the
tan of 112.5º, which is half of 225º. This cropped up because tan 45º = tan 225º, so either angle fits the equation, above).
BP is the tan of 22.5º times the
side AB, and the tan of 22.5º is sqrt(2)-1, so
BP=(sqrt(2)-1)(BC), so PC=(2-sqrt(2))(BC), which is exactly twice the
length of NO. | 