What is the area of a quadrilateral with a perimeter of 24 centimeters

Question

What is the area of a quadrilateral with a perimeter of 24 centimeters?

(1) The quadrilateral is formed by combining two isosceles right triangles
(2) All angles are of equal measure and the diagonals are perpendicular of each other.

chetan2u · Answer

ans B..
statement one can help us form different quadrilateral having different proportions of sides. one where the two triangles meet are joined by their hyp and other where hyp of one is superimposed on the base/perp of the other.. so not sufficient
statement 2 tells us it is a rectangle and the ratio of sides is given.. sufficient

bluesquare · Answer

1: The quadrilateral can either be a square or a quadrilateral with sides X, X, X root 2, and 2X. In the first case, the perimeter is 4X -> X is 6, so area is 36, and in the second case, the perimeter is 4X + X root 2 -> the area is not 36. So 1 is not sufficient.

2: All angles are equal -> quadrilateral is a rectangle, and the width is 25% of the length, so the sides are X, X, .25X and .25X -> perimeter is 2.5X and so the area is calculate-able.

Therefore, 2 is sufficient, so the answer is B.

Jackal · Answer

Quick approach

Statement 1: not possible as 24 needs to be divided into 2 pairs of equal numbers. Multiple possibilities. So out.

Statement 2: sufficient
 concept - sum of interior angles of a convex quadrilateral is always 360 degrees (assuming we have only convex polygons in gmat) 
Therefore quadrilateral is a rectangle. Its not a square as the relation between length and breadth is given. So solvable.

keats · Answer

This question is badly written.

1. How can perimeter be in square centimeters ?
2. Answers are contradictory through both the statements! A rectangle and a square. How is that possible ?

Jackal · Answer

totally agree on the framing of the question 1. Ignore square part in the units

2. But Square is a rectangle with all sides equal. please refer gmatclub math book.

keats · Answer

According to me the answer should be D.

You can easily see from statement 1 that it is a square.

a + b + c + d= 24
4a =6
a=1.5

Area = 2.25 square centimeters

Where as the second statement yields a different value of the area.

Please check. I was saying it in that terms :-D

Jackal · Answer

Hello. Good evening.

The statement 1 can be a square but always is not a square. Please refer attached diagram. As you rightly said this is not a well written problem. But in the Gmat world I think this is a smart problem.

keats · Answer

Oh Yes. Thanks a lot.

I tried forming the quadrilateral only by joining the two hypotenuses. But you are right in saying that it can be a square and cannot be a square. Well, you illustrated your point with a diagram that clears my doubt by joining the other triangle through the base of the first triangle. Yes, this is a smart problem and well an add on to my error log. Thank You Bunuel for the question :-D

UJs · Answer

What is the area of a quadrilateral with a perimeter of 24 square centimeters?

Stmt (1) :  The quadrilateral is formed by combining two isosceles right triangles

if both the triangles have same sides say  (1,1,2)  &  (1,1,2)  then it can be a rectangle.
but if one triangle is  (1,1,2) & (2, 2, 2 \sqrt{2} )  , then we can make one diagonal as 2 (2 as common side) and make a quadrilateral.

two different shapes    Not sufficient

Stmt (2) :  All angles are of equal measure and the width of the quadrilateral is 25% of the length
let angle of quadrilateral = a , then 4a = 360 , a = 90 (square or rectangle) , but length/width is mentioned so rectangle
now if length = x ; width = 25% of length = x/4

x + 0.25x = 24 ; can find length/width & area    sufficient

Ans   B

Bunuel · Answer

OFFICIAL SOLUTION: What is the area of a quadrilateral with a perimeter of 24 centimeters? (1) The quadrilateral is formed by combining two isosceles right triangles. It's easy to fall into the trap and consider only the case when the two isosceles right triangles give a square (figure 1) but there are other cases possible, for example, when two isosceles right triangles give a parallelogram (figure 2): Untitled.png So, this statement is not sufficient. (2) All angles are of equal measure and the diagonals are perpendicular of each other. The sum of interior angles of a quadrilateral is 360°. Thus, if all angles are of equal measure, then each angle must be 90°. So, we have a rectangle , with the diagonals which are perpendicular of each other. So, we have a square . Since the perimeter is 24, then 4x = 24. We can find x, hence we can find the area. Sufficient. Answer: B.

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What is the area of a quadrilateral with a perimeter of 24 centimeters

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