Hi,
There can be many circles drawn with same chord.
Example a circle can have this segment as one of its chords.Another can have this chord as a diameter.
So, there can be any number of values for the area of circle.
If the leg of a right triangle is fixed, say a units, then the perimeter has to be a+a+asqrt(2).

Hence, D

pankajks2010 wrote:If points X & Y are two distinct points in the co-ordinate plane, which of the following can have more than one possible value?

a) The circumference of a circle with diameter XY
b) The area of a square with diagonal XY
c) The perimeter of a right isosceles triangle with a leg XY
d) The area of a circle with chord XY
e) The area of an equilateral triangle with base XY

Source: Kaplan

OA is D, whereas, I got C

"Leg" is a term reserved for one of the sides forming the right angle. The third side (opposite the right angle) is known as the hypotenuse. If the line XY is one of the legs of a right isosceles triangle, that means that the other leg is also XY, and the hypotenuse will equal leg*√2. This 1:1:√2 ratio between the two equal legs and the hypotenuse in a right isosceles triangle is a fixed property of such triangles (it is derived from the Pythagorean theorem, which in a right isosceles triangles comes out as a^2 + a^2 = (hypotenuse)^2). In other words, in a right isoc. triangle, one side is enough to determine the other 2 sides, and thus the perimeter as well.

A Chord, however, is a line of unspecified length, as long as it is drawn between two points on the circle's circumference. A chord can be the diameter of the circle (if it is drawn between two points at 180 degrees from each other), but it can also be a shorter chord between two points that are closer to each other. Thus, without knowing exactly which chord XY represents in the circle, it is impossible to fix the area of the circle from the length of XY.

D it is,as chord xy will have min to max(=dia) lengths.Area of circles will differ hence.

C right isosceles will have hypotenuse ^2 = 2* xy^2. thus perimeter depends on XY value which is fixed.
POE.

Thanks all for the reply.

My confusion was with the definition of "leg" of an isosceles right triangle. I considered two options:
a) when XY is hypotenuse and b) when XY was non-hypotenuse. However, with Geva's detailed response, everything is clear now Thanks!!