DS - Geometry - Inscribed Angle

wilderness · Posted: Sun Jul 06, 2008 8:36 am Post subject: DS - Geometry - Inscribed Angle

Can anyone explain why ans is D ? Thanks

ed09 · Posted: Sun Jul 06, 2008 1:19 pm Post subject:

Hi,

My solution is this.

Evidently, triangles ABO and BOC are isosceles triangles, and CO=OB=AB=r.
Let's designate the angles in small letters for convenience.
BAO=BOA=a
ABO=b
CBO=BCO=c
BOC=d
COD=e

(1) c=40
b+c=180
b=180-c=180-40=140
a=(180-b)/2=(180-140)/2=20
SUFFICIENT

(2) e=60
a+d+e=180
a+d=180-e=180-60=120
a=(180-b)/2
c=(180-d)/2
b+c=180
b=180-c=180-(180-d)/2=(360-180+d)/2=(180+d)/2
a=(180-(180+d)/2)/2=(360-180-d)/4=(180-d)/4
a+d=((180-d)/4)+d=120 (see above)
180-d+4d=120*4
180+3d=480
3d=300
d=100
a=120-d=20
SUFFICIENT

The answer is D.

Best!

ed09 · Posted: Sun Jul 06, 2008 1:37 pm Post subject:

Sorry, I just mixed up the order of statements.
However, the sollution doesn't suffer.
Just read it in this order.

(1) e=60
a+d+e=180
a+d=180-e=180-60=120
a=(180-b)/2
c=(180-d)/2
b+c=180
b=180-c=180-(180-d)/2=(360-180+d)/2=(180+d)/2
a=(180-(180+d)/2)/2=(360-180-d)/4=(180-d)/4
a+d=((180-d)/4)+d=120 (see above)
180-d+4d=120*4
180+3d=480
3d=300
d=100
a=120-d=20
SUFFICIENT

(2) c=40
b+c=180
b=180-c=180-40=140
a=(180-b)/2=(180-140)/2=20
SUFFICIENT

The answer is D.

Besides, I am positive, that it's needless to solve this math, if you understand the concept and are sure that you can solve for the angle.

Best!