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# Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD

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Joined: 02 Sep 2009
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Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD  [#permalink]

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28 Mar 2019, 06:06
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Difficulty:

55% (hard)

Question Stats:

44% (02:51) correct 56% (03:04) wrong based on 18 sessions

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Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and triangle AED and triangle BEC have equal areas. What is AE?

(A) 9/2
(B) 50/11
(C) 21/4
(D) 17/3
(E) 6

Attachment:

d25c0c5968bc5ab64f9f91412801d2cdb764ce45.png [ 10.29 KiB | Viewed 696 times ]

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Re: Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD  [#permalink]

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28 Mar 2019, 12:39
chetan2u can you please explain this question?
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Re: Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD  [#permalink]

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28 Mar 2019, 12:51
1
Bunuel wrote:

Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and triangle AED and triangle BEC have equal areas. What is AE?

(A) 9/2
(B) 50/11
(C) 21/4
(D) 17/3
(E) 6

Attachment:
d25c0c5968bc5ab64f9f91412801d2cdb764ce45.png

∆ABE=∆DEC
AB/DC= 9/12 ; 3/4
so side EC in ∆DEC = 4 and side AE ∆ AEB= 3
AC ; 3:4 ratio ; 3/7 * 14; 6= AE
IMO E
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Re: Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD  [#permalink]

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28 Mar 2019, 19:34
1
Bunuel wrote:

Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and triangle AED and triangle BEC have equal areas. What is AE?

(A) 9/2
(B) 50/11
(C) 21/4
(D) 17/3
(E) 6

Attachment:
d25c0c5968bc5ab64f9f91412801d2cdb764ce45.png

AEB and CED are similar triangles since area of AED and BEC is equal. And Angle EAB=Angle CED(opposite angles)

Using similarity
Let AE=x, EC=14-x

$$\frac{x}{9}=\frac{14-x}{12}$$,

Solve for x, x=6.
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Re: Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD  [#permalink]

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28 Mar 2019, 21:03
Bunuel wrote:

Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and triangle AED and triangle BEC have equal areas. What is AE?

(A) 9/2
(B) 50/11
(C) 21/4
(D) 17/3
(E) 6

Attachment:
d25c0c5968bc5ab64f9f91412801d2cdb764ce45.png

Firstly, convex quadrilateral does not mean AB and CD are parallel, and so is wrong to take ABE and CDE similar without any reasoning. Convex means all angles are less than 180, and in GMAT a quadrilateral means all angles are less than 180.

YES, they are similar, but reasoning is-
Area of triangle AED and triangle BEC are equal. That is A(AED)=A(BEC), add A(ABE) to both sides. Thus, A(AED)+A(CDE)=A(BEC)+A(CDE), which becomes A(ACD)=A(BCD).
Both the triangles have same BASE, CD, so their height has to be EQUAL, and their height lies on AB, so both points on Ab should be equidistant from CD. THis is the reason why AB and CD are parallel.
Always write as per the equal angles, A=D, and so on.. so ABE~DCE so $$\frac{AB}{CD}=\frac{BE}{CE}$$= $$\frac{9}{12}=\frac{AC-CE}{CE}$$=$$\frac{3}{4}=\frac{14-CE}{CE}$$=3CE=56-4CE...7CE=56 or CE=8, and BE = 14-8=6

E
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Re: Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD   [#permalink] 28 Mar 2019, 21:03
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