Circle

[cypherskull]

For a circle with center point P, cord XY is the perpendicular bisector of radius AP (A is a point on the edge of the circle). What is the length of cord XY?

(1) The circumference of circle P is twice the area of circle P.

(2) The length of Arc XAY = 2pi/3.


The OA is D. I see how statement (1) is sufficient. But don't see how (2) will help since we don't know the radius. Please help!

[neelgandham]

From the diagram(attached), we know that PO = OA = AP/2 = r/2.
Length of line segment XO = √3/2*r (Pythagorean theorem)
We also know that XP = r

Sin(XOP) = r/2 / r = 1/2 = Sin 30. So angle XOP = 30 degrees and angle XPO = 60 degrees
We know that Angle XPY = 2* Angle XPO = 120 degrees
Frpm the question, length of Arc XAY = 2pi/3 = ((Angle of the arc)/360)*2*pi* radius of the circle
i.e. 2pi/3 = (120/360)*2*pi*r
i.e. r = 1
Length of Chord XY = 2* Length of line segment XO = 2* √3/2 * r = √3