chiragr wrote:
Jeff is painting two murals on the front of an old apartment building that he is renovating. One mural will cover the quadrilateral face ABCD while the other will cover the circular face (shown to the right, with radius XY). Assuming that the thickness of the coats of paint is negligible, will each mural require the same amount of paint? Note: Figures are not drawn to scale.
(1) AB = BC = CD = DA and \(AB=XY\sqrt{\pi}\)
(2) AC = BD and \(AC=XY\sqrt{2\pi}\)
We need to answer the question:
Is area_ABCD = pi(XY)^2 ?
Statement One Alone:=> AB = BC = CD = DA and AB = XY√pi
Squaring the second equation, we have:
AB^2 = pi(XY)^2
Using the above information, we can rephrase the question:
Is area_ABCD = AB^2 ?
Since AB = BC = CD = DA, the sides of the quadrilateral are equal.
If quadrilateral ABCD is a square, then the answer is Yes. Whereas, if ABCD is a rhombus but not a square, then the answer is No since area_ABCD would be less than AB^2. [Think of the area formula for a rhombus, area = base × height. If the interior angles on the base of the rhombus are not right angles, then the height of the rhombus is less than its side length.]
Statement one is not sufficient. Eliminate answer choices A and D.
Statement Two Alone:=> AC = BD and AC = XY√(2pi)
From the second equation, we have:
AC^2 = 2pi(XY)^2
(AC^2)/2 = pi(XY)^2
Using the above information, we can rephrase the question:
Is area_ABCD = (AC^2)/2 ?
Since AC = BD, the diagonals of the quadrilateral are equal.
If quadrilateral ABCD is a square, then the answer is Yes since a square’s area can be determined as (diagonal^2)/2. Whereas, if ABCD is a rectangle but not a square, then the answer is No since area_ABCD would be less than (AC^2)/2. [If a diagonal is given, then the square has the greatest area among all rectangles with the same diagonal.]
Statement two is not sufficient. Eliminate answer choice B.
Statements One And Two Together:Since the sides of the quadrilateral are equal and the diagonals of the quadrilateral are equal, quadrilateral ABCD must be a square. In this case, we have a definite Yes answer to the rephrased questions.
The two statements together are sufficient.
Answer: C