For a circle with center point P cord XY is the perpendicular bisector

since XY is perpendicular bisector of AP, we can draw a line from X to P and from Y to P, to form 2 30-60-90 triangles
why 30-60-90? because XP and YP is radius of the circle. AP is as well radius of the circle. Taking into consideration that XY bisects AP, we know for sure that:
from the intersection, say point Z, to P, we have 1/2 radius. we have 1 angle 90, and sides with proportions x-x sqrt 3 - 2x. this is a 30-60-90 triangle.

in order to find at least the radius.

from 1, we can deduct that r=2 - sufficient
from 2, we can deduct calculate the circumference, and get the radius - sufficient.

D

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.



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

(1) The circumference of circle P is twice the area of circle P.
(2) The length of Arc XAY = 2π3 .

If we let Z represent the intersection point between AP and XY, and ZP=r, XP=2r, XZ=sqrt(3)r, there is one variable (r), and 2 equations are given from the 2 conditions; there is high chance (D) will be our answer.
From condition 1, 2(phi)r=2(phi)r^2, r=1. This condition is sufficient,
From condition 2, angle AXP=120 deg, and the length of Arc XAY=2(phi)r(120/360)=2(phi)/3. r=1. This conditions is sufficient as well.
1)=2), so the answer becomes (D).

Hi Bunuel,
Thanks for the amazing explanation. But how did you figure out that AZ=ZP? Do we have to assume it from the figure because it is nowhere mentioned in the question. For that reason I chose B which was incorrect.

You should read the stem carefully. Check the highlighted part: bisector means that it cuts in half.

Hi Bunuel,

Thanks for the explanation. Sorry this maybe a very basic question - how do we know that XZ=ZY?

Here is a proof:

Bunuel bb, other moderators - need help here
In 2, how do we know r =1?
From the formula you used for arc length, we just get theta * r = 120.

I am having a hard time understanding how can we assume r = 1 here? it can be the angle subtended is 60 and the radius =2.

Thanks for the help

(2) says: "The length of Arc XAY = \(\frac{2\pi}{3}\)".

Arc XAY is 120 degrees, so its length is 1/3rd (120/360 = 1/3) of the whole circumference, which is \(2\pi{r}\). Thus, \(\frac{2\pi}{3}=\frac{120}{360}*2\pi{r}\). Solving this gives r = 1.

Can someone please explain that why 2πr is = 2*πr^2 ? .logic might be little bit behind but I am new with circle so please give hands to me here.

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From the diagram and the stem: AZ=ZP=r/2. How?

Have you read the whole discussion ? I think you must have missed this post.