Data Sufficiency

Can you answer using graph
In the xy-plane, point (r,s)lies on a circle with center at
the origin. What is the value of r^2 + s^2 ?
(1) The circle has radius 2.
(2) The point ( square root 2 and minus square root 2) lies on the circle.

To make the solution easier to follow, let's replace (r, s) with (a, b):

jkwan wrote:In the xy-plane, point (a, b) lies on a circle with center at the origin. What is the value of a² + b²?
(1) The circle has radius 2.
(2) The point (√2, -√2) lies on the circle.

The equation for a circle centered at the origin is x² + y² = r², where r is the radius.
Since (a, b) lies on the circle, we get:
a² + b² = r².
Thus, to determine the value of a² + b², we need to know the value of r.

Question rephrased:
What is the value of r?

Statement 1: The circle has radius 2.
SUFFICIENT.

Statement 2: The point (√2, -√2) lies on the circle.
Since the radius is equal to the distance between the origin and any point on the circle -- including (√2, -√2) -- the value of r can be determined.
SUFFICIENT.

The correct answer is D.

sajibfin06 wrote:Can you answer using graph
In the xy-plane, point (r,s)lies on a circle with center at
the origin. What is the value of r^2 + s^2 ?
(1) The circle has radius 2.
(2) The point ( square root 2 and minus square root 2) lies on the circle.

I was looking at this and realized that even though I don't really know the equation for a circle, a circle in the xy plane defines a whole bunch of right triangles with horizontal and vertical sides and with the radius of the circle as the hypotenuse.

The fact that the center of this circle is at the origin makes things pretty simple. The right triangles have one side length |x| and the other side length |y| and x^2 + y^2 = radius^2 for all x and y.

So r^2 + s^2 = radius^2 and in looking at Statement 1, I immediately see that r^2 = 4 and Statement 1 is sufficient.

Then Statement 2 provides x and y which we can use to to find x^2 + y^2 = radius^2. So Statement 2 is also sufficient.

Choose D.