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

Question

https://www.majortests.com/sat/testpics/p001-10.gif

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.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards