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555-605 (Medium)|   Geometry|      
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LordStark
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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = Updated on: 07 Oct 2018, 03:08
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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Difficulty:

35% (medium)

Question Stats:

79% (02:41) correct 21% (02:52) wrong based on 145 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

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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 06 Oct 2018, 23:35
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PeepalTree
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
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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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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 06 Oct 2018, 23:20
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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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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 06 Oct 2018, 23:24
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PeepalTree
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
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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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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = Updated on: 06 Oct 2018, 23:36
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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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)
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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 06 Oct 2018, 23:36
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src86
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)
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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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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 07 Oct 2018, 01:30
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chetan2u
PeepalTree
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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thank you chetan2u sorry for the typo
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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 07 Oct 2018, 07:24
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PeepalTree


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

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Attachment:
img1.jpg
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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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What is the area of the quadrilateral ABCE if BD = 8, AB = 6, and ED = 04 Aug 2020, 23:59
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LordStark


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

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img1.jpg
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Since triangles ABD & CDE are similar

and Since AD = 10 & ED = 5
EC = 3 and BC = 4

Area of quadrilateral ABCE = 1/2 (EC + AB)*BC = 1/2 (3 + 6) * 4 = 18

IMO E
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