Given a circle with diameter AB. Points C and D are selected on the circumference of the circle such that the chord CD intersects AB inside the circle, at point P. The ratio of the arc length AC to the arc length BD is 4:1, while the ratio of the arc length AD to the arc length BC is 3:2. Find ∠APC.
<APC = 45 deg
Start by calling arc_AC = 4x and arc_BD = x
Also arc_AD = 3y and arc_BC = 2y.
While x and y are unknown, any values will allow the arc ratios given.
The arcs on either side of the diameter AB add to 180
3y + x = 180 (arc_AD + arc_BD)
2y + 4x = 180 (arc_BC + arc_CA)
gives x=18, y=54
This gives AD=162, BD=18, CB=108, AC=72
Inscribed angles are half their intercepted arcs:
<CAB = 1/2 arc_CB = 54
<ACD = 1/2 arc_AD =81
So, for triangle APC, <APC = 180 - <CAP - <ACP
<APC = 180 - 54 - 81 = 45
QED
Edited on July 8, 2024, 2:16 pm
soln
