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Manager  Joined: 24 Aug 2012
Posts: 109
In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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5
13 00:00

Difficulty:   65% (hard)

Question Stats: 68% (02:55) correct 32% (03:02) wrong based on 266 sessions

### HideShow timer Statistics  In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ?

A. 18
B. 13.5
C. 9
D. 4.5
E. 3

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Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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3
1
kingb wrote: In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ?

A. 18
B. 13.5
C. 9
D. 4.5
E. 3

Attachment: Untitled.png [ 1.73 KiB | Viewed 32239 times ]

First of all notice that triangles CDE and BAE are similar (all three angles there are equal).

This means that the ratio of corresponding sides in these triangles are equal (corresponding sides are opposite equal angles): CD/AB = ED/AE --> 9/3 = ED/AE --> 3 = ED/AE --> 3*AE = ED.

Now, since AD = AE + ED = 4, then AE + 3*AE = 4 --> AE = 1 (and ED = 3).

The area of triangle AEC = 1/2*(base)*(height) = AE*CD = 1/2*1*9 = 4.5.

Hope it's clear.
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Schools: Cambridge'16
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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Made guess that 2<ED<4 according to drawing, so ED=3

Area ACD=18, Area ECD=13.5. So, area AEC=18-13.5=4.5

D
Manager  Joined: 09 Aug 2015
Posts: 84
GMAT 1: 770 Q51 V44 GPA: 2.3
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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Area of triangle ABC = 6.

6 = 1/2 * b * h where the base is 3, and the height is 4.

Now notice that AEC is more than half of ABC, therefore 4.5 is the answer.
Manager  Joined: 06 Mar 2014
Posts: 89
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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1
Hi Bunuel:

Can you please explain how is the height CD for triangle AEC.
Why is the height of triangle AEC CD??
Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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Shree9975 wrote:
Hi Bunuel:

Can you please explain how is the height CD for triangle AEC.
Why is the height of triangle AEC CD??

The altitude/height of a triangle is the perpendicular from the vertex to the opposite side. Note that the side may need to be extended. Thus CD is the height of AEC because CD is the perpendicular from vertex C to the line segment containing the opposite side AE.

Hope it's clear.
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Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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1
kingb wrote: In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ?

A. 18
B. 13.5
C. 9
D. 4.5
E. 3

Lay Down Foundation: Area of AEC = Area of ABC - Area of ABE

Also, Triangle ABE and CED are similar due to a pair of opposite angles and one right angle in each triangle

So AB/CD = AE/ED
i.e. 3/9 = AE/4-AE
i.e. AE = 1

So, Area of CED = [(1/2)*3*4 - (1/2)*3*1)] = 4.5

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Joined: 24 May 2016
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Location: Germany
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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Calculating the area for triangle ADC b*h/2 = 4*9/2= 18
From there you can eyeball and delete first of all option A. You see that triangle AEC is roughly one-third of triangle ADC, which would be 6.
The only option that comes close to that is option D.
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Posts: 11719
Re: In the figure above AD = 4, AB = 3 and CD = 9  [#permalink]

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_________________ Re: In the figure above AD = 4, AB = 3 and CD = 9   [#permalink] 02 Sep 2018, 09:45
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