If points A, B and C lie on a circle with center O, what is the area

Question

If points A, B and C lie on a circle with center O, what is the area of the circle?

(1) Line BC has a length of 4.

(2) The length of line BC is 1/2 the length of AC.

PKN · Answer

Question stem:-what is the area of the circle? (We require radius or diameter)

St1:- Line BC has a length of 4.
We don't know whether BC is the radius or diameter of the circle.
Hence, we can't determine area of the circle.
Insufficient.

St2:-The length of line BC is 1/2 the length of AC.
Implies that 
a) There are numerous points on the circumference that meets the given criteria
b) BCA is a semi-circle with AB as the diameter.

As we don't know the positions of A,B, and C, we can;t determine the radius, Subsequently failed to determine area of circle.

Insufficient.

Combined, We know, BC=4, So, AC=2*BC=2*4=8.
 case(a) above isn't valid as the the measure of BC & AC is fixed.
With case(b), ABC is a semicircle, so ABC is a right angled triangle with AB=diameter of circle=hypotenuse of triangle, BC=4 and AC=8
So,  2r=AB=\sqrt{8^2+4^2} 
Hence, radius can be determined.Subsequently, area of circle can be determined. 
 Sufficient.
Ans. (C) 
 This is one possibility out of numerous possibilities.

Rectifying my mistake 
Combined, We can draw several circles with BC=4 cm and AC=8 cm with different measure of diameters. 
No unique value of radius. So, area of circle has no unique value.
Therefore, insufficient.
Ans. (E)
Enclosure:- Diagram
P.S:- Earlier I have considered only one possibility.

US09 · Answer

Hi,

Can you please elaborate how you deduced from statement 2 that BCA is a semicircle. I don't understand what I am missing here.

Thank you!

PKN · Answer

Hi  urvashis09 ,
As I mentioned, we have fixed measure of BC & AC, i.e, 4 and 8 respectively.
Our aim is to find radius or diameter. We have to explore the possibilities of finding out radius or dia.
a) You know when the side of the triangle will become the diameter, then we would achieve our target. How the side of a triangle will become the diameter ? It is only when the vertices of the triangle lie on the semicircle.
b) Given two sides of a triangle with different measure of length , a fixed measure of the 3rd side can only be determined when we make the triangle a right angled one.

This is how I proceeded.

PS:- In st2:- (a) and (b) are the possibilities, not necessarily true.

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If points A, B and C lie on a circle with center O, what is the area

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