In the figure above AD = 4, AB = 3 and CD = 9

Question

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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

Bunuel · Accepted Answer

Untitled.png 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. Answer: D. Hope it's clear.

Temurkhon · Answer

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

ggurface · Answer

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.

Shree9975 · Answer

Hi  Bunuel :

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

Bunuel · Answer

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.

GMATinsight · Answer

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 =   = 4.5

Answer: Option D

AK92 · Answer

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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In the figure above AD = 4, AB = 3 and CD = 9

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In the figure above AD = 4, AB = 3 and CD = 9

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