Circles DS

[shankar.ashwin]

In a circle with center O, cord XY is the perpendicular bisector of radius AO (A is a point on the edge of the circle). What is the length of cord XY?


(1) The circumference of circle O is twice the area of circle O.
(2) Length of arc XAY is = 2Pi / 3.

[cans]

let radius =r.
and XY bisects AO at B.
BO = r/2
also in triangle XOB, XB is perpendicular to OB.
sin OXB = (r/2)/r = 1/2
thus OXB = 30 and thus XB = root(3) * r/2
and XY = root(3)*r
Thus we need to find r.

A) 2 pi r = 2* pi r r
or r=1 sufficient

B) XY subtends 120 degree at centre. thus length = 2pi r /3 = 2 pi/3 or r=1
Sufficient
IMO D

[bijoyajj]

B) XY subtends 120 degree at centre. thus length = 2pi r /3 = 2 pi/3 or r=1
Sufficient
IMO D

How you knew the angle was 120 degree without knowing the radius?..

P.S:- I am yet to prepare for geometry.. pardon me if this is some straight forward rule.. i can make out that length of an arc can be figured from circumfrence and the angle it make..

[shankar.ashwin]

If you could follow the first part of Cans explanation, you know triangle XOB is a 30-60-90 triangle as the sides are in a ration 1: sqrt3 : 2

Angle X is thus 30, you know OX = OY (radius)

So consider triangle OXY which is a 30-30-120 triangle.

And hence the angle is 120.

OA is D

B) XY subtends 120 degree at centre. thus length = 2pi r /3 = 2 pi/3 or r=1
Sufficient
IMO D

How you knew the angle was 120 degree without knowing the radius?..

P.S:- I am yet to prepare for geometry.. pardon me if this is some straight forward rule.. i can make out that length of an arc can be figured from circumfrence and the angle it make..