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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =

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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post Updated on: 07 Oct 2018, 03:08
1
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

76% (02:21) correct 24% (02:58) wrong based on 41 sessions

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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 5? (Note: Figure not drawn to scale.)

A. 10
B. 12
C. 14
D. 16
E. 18


Source:800 score

Attachment:
img1.jpg
img1.jpg [ 4.08 KiB | Viewed 1015 times ]

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Originally posted by LordStark on 06 Oct 2018, 22:21.
Last edited by Bunuel on 07 Oct 2018, 03:08, edited 3 times in total.
Renamed the topic and edited the question.
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 06 Oct 2018, 23:35
2
PeepalTree wrote:
What is the area of the quadrilateral ABCE if BD=8, AB=6, and ED=5?

a. 10
b. 12
c. 14
d. 16
e. 18

Source:800 score


Assuming we are looking for area of quad ABCE,

Triangle ABD is right angled. AB = 6 and BD = 8 so using multiple of pythagorean triplet 3-4-5, AD = 10

Triangles ABD and ECD are similar by AA (both have a common angle D and 90 degree angles)
So
\(\frac{Area ECD}{Area ABD} = (\frac{ED}{AD})^2 = \frac{1}{4}\)

Area ABCE = (3/4)*Area of triangle ABD = (3/4) * (1/2) * 6 * 8 = 18

Answer (E)
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 06 Oct 2018, 23:24
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2
PeepalTree wrote:
What is the area of the quadrilateral ABCD if BD=8, AB=6, and ED=5?

a. 10
b. 12
c. 14
d. 16
e. 18

Source:800 score



I am sure PeepalTree, you mean ABCE and not ABCD..
Now let's solve step wise..

Step I....
Take ∆ABD, a right angled triangle with sides in ratio 3:4:5 so AD = 2*5=10 or
\(AD^2=AB^2+BD^2=6^2+8^2=10^2......AD=10\)
Step II....
Take ∆s ABD and DEC..
AB and CE are perpendicular on BD..
So ABD and DEC are similar ..
Now ED:AD =5:10=1:2
So areas will be in ratio 1^2:2^2=1:4
Step III....
Now area of ∆ABD is AB*BD*1/2=8*6/2=24
Area of ∆DEC is 24*1/4=6.. rule of ratios of similar triangles
Area of quadrilateral / trapezium ABCE is area of ∆ABD-area of ∆DCE=24-6=18

E
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 06 Oct 2018, 23:36
1
src86 wrote:
Hi Chetan,
From Triangle ABD, AE = 5.
And since opp. angles in the quadrilateral are 90 degree each, I took AB=FC = 6 and E as the mid pt. (based on the figure). Is this incorrect approach ?

Area of Quadrilateral ABCF = 6x4 =24 sq. units. and area of Triangle AEF = 6 sq. units.

Therefore, area of quad. ABCE = 24 -6 = 18 (E)


No, you cannot take E as the midpoint unless it is clearly specified or you prove and then work ahead.
But a big NO to taking these estimates from the figures
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 06 Oct 2018, 23:20
Hi,
Are you sure the question is asking for the area of the quadrilateral ABCD ? Because it is not making any sense from the given diagram ? Correct me if I my understanding is wrong.

//Sayan
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post Updated on: 06 Oct 2018, 23:36
Hi Chetan,
From Triangle ABD, AE = 5.
And since opp. angles in the quadrilateral are 90 degree each, I took AB=FC = 6 and E as the mid pt. (based on the figure). Is this incorrect approach ?

Also I figured Triangle ABD and DEC are similar,

CE/AB = DE/AD

Therefore CE =3 and it satisfies the mid pt. assumption.

Area of Quadrilateral ABCF = 6x4 =24 sq. units. and area of Triangle AEF = 6 sq. units.

Therefore, area of quad. ABCE = 24 -6 = 18 (E)

Originally posted by src86 on 06 Oct 2018, 23:32.
Last edited by src86 on 06 Oct 2018, 23:36, edited 1 time in total.
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 07 Oct 2018, 01:30
chetan2u wrote:
PeepalTree wrote:
What is the area of the quadrilateral ABCD if BD=8, AB=6, and ED=5?

a. 10
b. 12
c. 14
d. 16
e. 18

Source:800 score



I am sure PeepalTree, you mean ABCE and not ABCD..
Now let's solve step wise..

Step I....
Take ∆ABD, a right angled triangle with sides in ratio 3:4:5 so AD = 2*5=10 or
\(AD^2=AB^2+BD^2=6^2+8^2=10^2......AD=10\)
Step II....
Take ∆s ABD and DEC..
AB and CE are perpendicular on BD..
So ABD and DEC are similar ..
Now ED:AD =5:10=1:2
So areas will be in ratio 1^2:2^2=1:4
Step III....
Now area of ∆ABD is AB*BD*1/2=8*6/2=24
Area of ∆DEC is 24*1/4=6.. rule of ratios of similar triangles
Area of quadrilateral / trapezium ABCE is area of ∆ABD-area of ∆DCE=24-6=18

E


thank you chetan2u sorry for the typo
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =  [#permalink]

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New post 07 Oct 2018, 07:24
PeepalTree wrote:
Image

What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 5? (Note: Figure not drawn to scale.)

A. 10
B. 12
C. 14
D. 16
E. 18


Source:800 score

Attachment:
img1.jpg



Area of rectangle = width * length = 4*6=24

Area of Right tringle = 3*4*1/2 = 6

Area of quadrilateral = 24-6 = 18
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Re: What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED =   [#permalink] 07 Oct 2018, 07:24
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