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

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sugu86
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ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink] New post   Updated on: 24 Feb 2018, 00:41
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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

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2014-11-18_2051.png
2014-11-18_2051.png [ 8.02 KiB | Viewed 20594 times ]
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D

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] New post   26 Apr 2014, 11:18
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time for basics..sometimes it really helps.. :lol: .https://www.cliffsnotes.com/math/geometr ... rence-area
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area formula.jpg
area formula.jpg [ 45.3 KiB | Viewed 18430 times ]

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



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] 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] 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] 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 17354 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] 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] New post   02 Sep 2014, 01:24
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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] 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] New post   18 Nov 2014, 09:55
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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



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:

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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink] 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] 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] New post   08 Sep 2017, 08:12
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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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ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink] New post   20 Nov 2018, 13:15
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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 [D]
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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink] New post   21 Apr 2020, 13:11
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



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

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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Re: ABCD is a parallelogram (see figure). The ratio of DE to EC [#permalink] New post   17 May 2023, 09:05
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