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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 [#permalink]

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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

Attachment:
2014-11-18_2051.png
2014-11-18_2051.png [ 8.02 KiB | Viewed 7742 times ]

Originally posted by sugu86 on 26 Apr 2012, 00:57.
Last edited by Bunuel on 24 Feb 2018, 00:41, edited 4 times in total.
Edited the question and the figure
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 26 Apr 2012, 01:47
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sugu86 wrote:
ABCD (attached as a jpg file ) is a paralleogram. 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
Thanks,

Suganth


Image
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.
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 26 Apr 2014, 11:18
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1
time for basics..sometimes it really helps.. :lol: .http://www.cliffsnotes.com/math/geometr ... rence-area
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area formula.jpg
area formula.jpg [ 45.3 KiB | Viewed 7129 times ]

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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 25 Aug 2014, 19:11
Bunuel wrote:
sugu86 wrote:
ABCD (attached as a jpg file ) is a paralleogram. 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
Thanks,

Suganth


Attachment:
ABCD.png
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.

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?
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 25 Aug 2014, 21:07
sagnik2422 wrote:
Bunuel wrote:
Attachment:
ABCD.png
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 -->

\(the area ABCE=\frac{(4x+3x)}{2}*3=21\) --> 7x=14 --> x=2;

The area of ABCD=12x=24

Answer: D.


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?




Area of Trapezoid is Avg of Parallel Sides \(\frac{(4x+3x)}{2}\)*Height...\(\frac{7x}{2}*3=21\)
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 26 Aug 2014, 00:56
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Given that area of shaded region = 21

Attachment:
ABCD.png
ABCD.png [ 4.96 KiB | Viewed 6572 times ]

Setting up the equation

\(9x + \frac{3x}{2} = 21\)

x = 2

Area of ABCD = 18 + 3 + 3 = 24

Answer = D
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 28 Aug 2014, 04:00
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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.
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 02 Sep 2014, 01:24
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
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 18 Nov 2014, 09:17
Bunuel wrote:
sugu86 wrote:
ABCD (attached as a jpg file ) is a paralleogram. 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
Thanks,

Suganth


Attachment:
ABCD.png
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.


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.
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 18 Nov 2014, 09:55
Ankur9 wrote:
Bunuel wrote:
sugu86 wrote:
ABCD (attached as a jpg file ) is a paralleogram. 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
Thanks,

Suganth


Image
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.


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.



Original image indicates that AE is perpendicular to CD:

Image
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 11 Jan 2015, 06:12
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?
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 27 Apr 2016, 18:01
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.
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink]

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New post 08 Sep 2017, 08:12
sugu86 wrote:
Attachment:
2014-11-18_2051.png

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


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 \(\triangle ADE\)
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 \(ABCD = 4xh = 4*6 = 24\)
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC   [#permalink] 08 Sep 2017, 08:12
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