ABCD is a parallelogram (see figure). The ratio of DE to EC

Question

https://gmatclub.com/forum/download/file.php?id=24780 ABCD is a parallelogram (see figure). The ratio of DE to EC is 1:3. AE has a length of 3. If quadrilateral ABCE has an area of 21, what is the area of ABCD? A. 18 B. 20 C. 22 D. 24 E. 26 2014-11-18_2051.png

hangovrlyf · Accepted Answer

time for basics..sometimes it really helps.. :lol:

.https://www.cliffsnotes.com/math/geometr ... rence-area

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=24780 
 ABCD is a parallelogram (see figure). The ratio of DE to EC is 1:3. AE has a length of 3. If quadrilateral ABCE has an area of 21, what is the area of ABCD?

A. 18 
B. 20 
C. 22 
D. 24 
E. 26

The area of a parallelogram is base*height --> the area of ABCD=(x+3x)*3=12x;

Now, notice that quadrilateral ABCE is a trapezoid. The area of a trapezoid is the average base length times altitude --> the area ABCE=(4x+3x)/2*3=21 --> 7x=14 --> x=2;

The area of ABCD=12x=24

Answer: D.

sagnik2422 · Answer

I am confused on how you got from ABCE=(4x+3x)/2*3=21 --> 7x=14 --> x=2; isn't 4x + 3x = 7x, and this is divided by 6, multiply both sides by 6 get 7x = 126, x = 18?

WoundedTiger · Answer

Area of Trapezoid is Avg of Parallel Sides  \frac{(4x+3x)}{2} *Height... \frac{7x}{2}*3=21

PareshGmat · Answer

Given that area of shaded region = 21 ABCD.png Setting up the equation 9x + \frac{3x}{2} = 21 x = 2 Area of ABCD = 18 + 3 + 3 = 24 Answer = D

bazc · Answer

Alternative simpler approach:

Note that the pair of triangles in the parallelogram if placed adjacent adjoining it's hypotenuse - will become a rectangle with the height as length and base as breadth.

So try to imagine, that this parallelogram is made of 8 equal triangles. (Triangle ADE + 3 pairs in between + one to the right)

We are given area of 7 triangles = 21

We are asked, area of 8 triangles = 24.

Sorry I am unable to draw the triangles on my phone to explain. Hope his helps.

divyakp · Answer

area of trapezoid ABCE = 21
(b1+b2)*h/2 
(3x + 4x)*3/2
x = 2
area of ABCD //gm = b*h
8*3=24.

answer is D

Ankur9 · Answer

Hi Bunuel,

How do we make sure whether AE is perpendicular to CD as it is not stated explicitly and we cannot assume it as height.
I solved the same way as you did but my first thought was that one cannot assume AE as height.
Thanks.

Bunuel · Answer

Original image indicates that AE is perpendicular to CD:

https://gmatclub.com/forum/download/file.php?id=24780

pacifist85 · Answer

I created another perpendicular from angle C to side AB, which created another right triangle, same as the one already created.

Then I added the area of the parallelogram (I named the point where the perpendicular line from angle C crosses AB as "f") that was created if you take the 2 trinagles out and the area of one of the triangles. So, I ended up with:
Aafce + Afbc = 21
21 = b*h + (b*h)/2
21 = 3x(3) + (3x/2)
21 = 9x + (3x/2)
21 = (18 + 3x) / 2
42 = 21x
x = 42/21
x = 2

So, now I have the value f x and can calculate the rest:

Aabcd = b * h = 8 * 3 = 24.

That's also right, right?

lalania1 · Answer

ABCD is a parallelogram. The area of a parallelogram is B * H => (X+3X) (3) = 12x
ABCE is a trapezoid. the area of a trapezoid is ((b1 + b2)/2) * h => ((4x+3x)/2) * 3 = 21 => x =2

if x =2 => 12x =Area of parallelogram => 12(2) = 24.

jedit · Answer

Let h be the height.

Area of  ABCD = 4x*h 
Looking at the picture, Area of Quadrilateral  ABCE  is Area of  ABCD  minus area of the  	riangle ADE 
Area of Quadrilateral  ABCE = 4x*h - \frac{1}{2}*xh = 21

xh*(4-\frac{1}{2}) = 21

xh*\frac{7}{2} = 21 
 xh = 21*\frac{2}{7} = 6

Area of  ABCD = 4xh = 4*6 = 24

keatross · Answer

Using logic we can solve fairly quickly:

Using the relationship between DE & EC, and the length of AE we can determine the area of the rectangle in AECB.

if
x=1
EC = 3
square area = 3x3 = 9 ... too small

if
x=2
EC = 6
square area = 6x3 = 18 ... just right

if
x=3
EC = 9
square area = 9x3 = 27 ... too big

Using the information, paired with what we just calculated, we can determine that each right angle triangle on each side of the rectangle has an area of 3.

Therefore, 18+3+3 = 24

jabhatta2 · Answer

Hi  Bunuel

If i am aiming for 700 + and i got this problem accurate but it took me 2.20 seconds -- does this mean i have to do this problem again (as part of my error log) even though i did eventually get it right

Reason for delay : by the time i took to understand the problem and write down and actually get D -- it took me 2 mins or so

what are your thought on doing a problem like this in 2.20 seconds ?

is that too long in your view

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ABCD is a parallelogram (see figure). The ratio of DE to EC

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ABCD is a parallelogram (see figure). The ratio of DE to EC

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