point P is in a regular octagon ABCDEFGH so that triangle ABP is equilateral. how many degrees in angle APC?
- Evolving SquidLv 61 decade agoFavorite Answer
The angle APB is 60 degrees and the triangle is equilateral, therefore angle ABP must be 60 degrees. By definition, the angle ABC must be 135 degrees for an octagon, so the angle CBP must be 75 degrees.
Points C, B, and P form an isosceles triangle with angle CBP equal to 75 degrees. By definition, angles BCP and CPB must be equal to half of (180-CBP), or 52.5 degrees.
Here is a diagram:
Angle APC is the sum of angles APB and CPB, or 60+52.5 or 112.5 degrees.
- Iby KLv 71 decade ago
if it is regular octagon, APB can't be equilateral (for this all three angles would have to be 60deg) - it is isosceles.
360deg/8=45deg which is APB angle (from A to B) in a regular octagon.
APC is just double angle (from A to C) of that of APB. result is 90deg.
- Jerome JLv 71 decade ago
m<ABC = 180º - 360º/8 = 135º
m<PBC = 135º - 60º = 75º
m<APB = m<ABP = m<BAP = 60º
Let s = AB ...... one side of octagon
AP = BP = BC = s
m<BPC = m<BCP ................ because BP = BC
m<BPC + m<BCP + m<PBC = 180º......... triangle BPC
m<BPC + m<BCP + 75 = 180º
m<BPC + m<BCP = 105º
2m<BPC = 105º
m<BPC = 52.5º
m<APC = m<APB + m<BPC = 60º + 52.5º = 112.5º