Points X, Y, and Z lie on circle C, and line segment XY pass

Question

Points X, Y, and Z lie on circle C, and line segment XY passes through the center of the circle. If the area of circle C is 18pi, what is the greatest possible perimeter of triangle XYZ?

(A) 18
(B) 3r2 + 18
(C) 6r2 + 6
(D) 6r2 + 12
(E) 9r2

GH-04.08.13 | OE to follow

Chiranjeevee · Answer

Lets O be the center of the circle. From point O, draw a line perpendicular to XY such that it will intersect the circle at Z. That will be the greatest possible perimeter of triangle XYZ.

Area of the circle = (pi)r^2 = 18pi

so, r = 3sqrt2 = OX = OY = OZ (all are radius of the circle)
YZ^2 = OY^2 + OZ^2 = (3sqrt)^2 + (3sqrt)^2 = 36
YZ = 6

diameter, XY= 6sqrt2
XZ = YZ = 6
so perimeter = 6 + 6 + 6sqrt2 = 12 + 6sqrt2. hence D.

Bunuel · Answer

The area of circle C is 18pi -->  \pi{r^2}=18\pi  -->  r=3\sqrt{2}  --> the diameter = XY =  6\sqrt{2} .

XY passes through the center of the circle implies that triangle XYZ is a right triangle and XY is its hypotenuse ( a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle ).

To maximize perimeter triangle must be isosceles, thus 45-45-90, with ratio of sides  1:1:\sqrt{2} . This means that its legs must be 6 and 6.

Perimeter  6+6+6\sqrt{2}=12+6\sqrt{2} .

Answer: D.

avohden · Answer

Official Explanation

Answer: D    The triangle with the greatest possible perimeter will be an isosceles 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).

ronr34 · Answer

Why is triangle 1:1:sqrt2 the one with the largest perimeter?

avohden · Answer

Please refer to the official explanation to answer this question. Excerpted below.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Points X, Y, and Z lie on circle C, and line segment XY pass

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Points X, Y, and Z lie on circle C, and line segment XY pass

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards