The triangle with the greatest possible perimeter will be an isoceles triangle with the diameter (XY) as a base. To understand why, think about the perimeter of very "skinny" triangles. These perimeters will be just barely greater than twice the diameter. As you make the triangle "fatter," the longer of the two sides gets only a little bit shorter, but the short side gets much longer.

That means that Z is halfway between X and Y on the circle. XY is a diameter, and Z is a radius perpendicular to that diameter. Before going any farther, figure out what the radius and diameter is, from the area given:

a = (pi)r^2 = 18pi

r^2 = 18

r = 3r2

d = 6r2

This tells us that XY = 6r2 and CZ is 3r2. The easiest way to find XZ and YZ is to think of triangle XYZ and two isoceles right triangles. For instance, in XCZ, XC is a radius and CZ is a radius. Both of the legs are 3r2 and they join at a right angle, so we can use the 45:45:90 triangle ratio. The hypotenuse is r2 times the length of a leg, so the length of the hypotenuse (XZ) is (3r2)(r2) = 6. The same reasoning indicates that YZ is 6 as well.

The perimeter, then, is the sum of XY, XZ, and YZ: 6r2 + 6 + 6 = 6r2 + 12, choice (D).