In the diagram shown below, ABCD is a rectangle. X and Y are points on

Question

In the diagram shown below, ABCD is a rectangle. X and Y are points on BC and AB such that Area of triangle AYD = Area of triangle DXC = 4 and Area of triangle BXY = 9. What is the area of the rectangle? 2.jpg A) 18 B) 22 C) 24 D) 26 E) 32

itsprateek95 · Answer

IMO E (32). Not the exact, butt fast method using options.
L= length B = Breadth
AY = L-y; YB = y
CX = x ;BX = B-x

ar (AYD) = 0.5*AY*AD ---> (L-y)B = 8 ......(1)

ar(BXY) = 0.5*BY*BX ----> y(B-x) = 18 ....(2)

Adding 1 and 2,

LB(Area of rectangle) = 26 + xy

As x,y is positive, Area = 32(only option greater than 26)

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GMATWhizTeam · Answer

Area of AYD = Area of DXC = 4. 
 \frac{1}{2} * AY * AD = 4 
 AY * b = 8 . .........(i)
 \frac{1}{2} * XC * DC = 4 
 XC * l = 8 .........(ii)

Area of BXY = 9. 
 \frac{1}{2} * BY * BX = 9  
 BY * BX = 18 
 (l-AY)*(b-XC) = 18 
 (l-\frac{8}{b}) * (b-\frac{8}{l}) = 18 
 lb-\frac{64}{lb}=18+16 
 (lb)^2-34lb+64=0.  
 (lb)^-2lb-32lb+64=0 
 lb(lb-2)-32(lb-2)=0 
 (lb-2)(lb-32)=0 
 lb = 2  or  32.  
 lb = 32   according to the options.

hence the right answer is Option E.

Asitima · Answer

I got the answer through this approach, which I am not sure is a good approach even though I got the answer correct. Experts can help?

1) AY*AD=8=CX*CD ----I
2) BY*BX=18----II
3) Length of ABCD = AD, Width of ABCD = CD (only for ref)

From equation I, we know {AY,AD} and {CX,CD} are different subsets of {1,2,4,8} as {1,8} or {2,4} 
 The big assumption here is that it is a rectangle and not square, hence both subsets cannot be same.

Taking Case 1: AY = 1, AD = 8 and CX = 2, CD = 4 
We have AD*CD= 8*4 = 32

Taking Case 2: AY = 2, AD = 4 and CX = 1, CD = 8
We have AD*CD= 32

So answer choice E.

I want to know if this assumption I took is fair, therefore is this a valid approach?
Thanks!

Sushen · Answer

Another method:
First, I wrote down the equations for the areas of the triangles and the rectangle.
I renamed the line segments to make it easier:
AY=a
AD=b
DC=c
CX=d
XB=e
BY=f
QUANTITY NEEDED = bc
1/2 ab = 4
1/2 cd = 4
1/2 ef = 9
b=8/a
c=8/d
bc = 64/ad
i.e. bc = a factor of 64 (... or 2^n)
The only answer that is a factor of 64 is 32! E and out!

Fdambro294 · Answer

Without any algebra, you can notice that the Areas given are Integer results and all the answer choices for the Rectangle’s Area are also integer results. It is therefore likely that the dimensions of each given line segment length is an Integer.

Area of triangle ADY = area of triangle DXC = 4

(1/2) L W = 4

L * W = 8

Since the other triangle, BXY, has a different integer area of 9, it seems likely that the rectangle is not a square and adjacent sides AD and DC are different Positive Integer lengths.

2 positive integers that multiple to a product or 8 are:

4 * 2 = 8 ————— and ———— 1 * 8 = 8

Case 1: let AD = 8 and DC = 4

To keep the area of the 2 right triangles DAY and DCX at 4 ——-> this would mean AY = 1 and XC = 2

Because it is a rectangle and the opposite sides are equal:

AY + YB = DC ——-> 1 + YB = 4 ———> YB = 3

And

CX + XB = AD ———-> 2 + XB = 8———> XB = 6

Area of right triangle BXY would be = (1/2) (YB) (XB) = (1/2) (3) (6) = 9

Area of triangle AYD = (1/2) (DA) (AY) = (1/2) (8) (1) = 4

And area of triangle DCX = (1/2) (DC) (CX) = (1/2) (4) (2) = 4

All the Triangle Area constraints are satisfied and the rectangle’s dimensions are given by:

DC = AB = 4 ——— and ———— AD = BC = 8

****(in fact, if you switched the dimensions around, you would still come up with a valid rectangle that satisfies the triangle area constraints with DC = AB = 8 and AD = BC = 4......since we are only concerned with the Area of the rectangle and not which side is the length and which side is the width, you still end up with the Same Answer)****

Area of Rectangle = (4) (8) = 32

32

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In the diagram shown below, ABCD is a rectangle. X and Y are points on

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