O is the center of the circle above, OB=2, and angle AOB

Question

https://gmatclub.com/forum/download/file2.php?mode=view&id=20793 O is the center of the circle above, OB = 2, and angle AOB measures 120 degrees. What is the area of the triangular region AOB? A. 1 B. 2 C. \frac{\sqrt{3}}{2} D. \sqrt{3} E. 2\sqrt{3} Source: GMAT Prep Question Pack 1 Difficulty: Hard Screen Shot 2013-08-17 at 2.24.00 PM.png

blueseas · Accepted Answer

SINCE ANGLE AT CENTRE =  120 
when you draw perpendicular OC on AB
it divides in two equal  30 - 60 - 90  triangle see figure
sides are in ratio of 1: \sqrt{3}: 2 
therefore OC =  1  and  CA = CB = \sqrt{3} 
THEREFORE AB =  2\sqrt{3} 
Hence AREA Of triangle =  1/2*base*height 
 =  1/2 * 2\sqrt{3}*1 = \sqrt{3} 
hence D

Chiranjeevee · Answer

Here radius is the side of the triangle. So, Area of the traingle = (1/2)*r*r*sin(angle between those two sides)
so area = (1/2)*2*2*sin120 = (1/2)*2*2*sin60 = sqrt3.

nsaruna · Answer

Extend OA to a point on the circle as point D. hence DAB is a right 30-60-90 triangle. Sin 60 will give AB to be 2sqrt3. Hence Area will be 1/2 * ((2sqrt3)/2) * ( sqrt(4-3)) = sqrt3

Konstantin1983 · Answer

Please move this question to PS section!

Bunuel · Answer

Done. Thank you for noticing.

keats · Answer

Area of triangle = 0.5*a*b*sin(theta)
= 0.5*2*2*sin(120)
= 2*sin(90+30)
= 2*cos 30
= 2*(1.732/2)
= root(3)

T700ISB · Answer

Area = 1/2 AB Sin(included angle)
 = 1/2 x 4 x sin(180-60) 
 =1/2 x 4 x sin 60
 = Rt 3.

nightblade354 · Answer

I have a question about this problem. Because AOB is 120 degrees, doesn't that mean that the side across AOB (the base) is the largest side? In the problem, angle A should be smaller than angle O. A = 2, so O should be greater than 2. Given this, if we take 1/2 base * height, we get 2 * (at least 2)*1/2, which gives us an area of at least 2. Yet, the answer of root 3 is 1.7. How is this possible?

BrentGMATPrepNow · Answer

First, since OA and OB are radii, they both have the same length (length 2)
This means ∆AOB is an ISOSCELES triangle, which means the other 2 angles are 30º each. 
 https://s30.postimg.cc/xanlfb1v1/AOB_1.jpg

Once I see the 30º angles, I start thinking of the "special" 30-60-90 right triangle, which we know a lot about. If we draw a line from point O so that it's PERPENDICULAR to AB, we can see that we have two 30-60-90 right triangles hiding within this diagram. PERFECT!
 https://s30.postimg.cc/9l2r4csvh/AOB_2.jpg

I've added (in purple) a 30-60-90 right triangle with measurements. 
We can use this to determine the lengths of some important sides. 
 https://s30.postimg.cc/xrdefhezx/AOB_3.jpg

At this point, we can see that our original triangle has a base with length  2√3  and height  1 
 https://s30.postimg.cc/6s9jkbsj1/AOB_4.jpg

Area = (base)(height)/2
So, the area of ∆AOB = ( 2√3 )( 1 )/2
= √3
= D

Cheers,
Brent

Basshead · Answer

Since AOB measures 120 degrees, we can split triangle AOB into two 30:60:90 right triangles.

The ratio of a  30:60:90  triangle is  1:\sqrt{3}:2

Since we are told the hypotenuse is 2, we have the following ratios:  1:\sqrt{3}:2

\frac{1}{2}b * h * 2 = \frac{\sqrt{3}}{2} * 2 = \sqrt{3}

Answer is 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.

O is the center of the circle above, OB=2, and angle AOB

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

O is the center of the circle above, OB=2, and angle AOB

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