Triangles and Semicircles

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masuarezdl: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Thu Jan 29, 2009 9:57 am; Thanked: 1 times

by masuarezdl » Sat Jan 31, 2009 12:16 pm

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60°.
(2) The degree measure of angle BCO is 40°.

I certainly dont see a way to solve this... Anybody?

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Source: Beat The GMAT — Data Sufficiency | Sat Jan 31, 2009 12:16 pm

vikram_k51: Master | Next Rank: 500 Posts; Posts: 208; Joined: Sat Jan 31, 2009 11:32 am; Location: Mumbai; Thanked: 2 times

Re: Triangles and Semicircles

by vikram_k51 » Sat Jan 31, 2009 12:32 pm

--------------------------------------------------------------------------------

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60°.
(2) The degree measure of angle BCO is 40°.

I guess the answer is C.

From the question:

AB=OC=OB(Since OC=OB=radius)

<BAO=<BOA(ABC is isoceles)

OBC=OCB(ISoceles)

Thus <OCB=<OBC

EAch alone is not sufficient.

Now taking A & B together

<BOC=180-80=100

<BOA=20=<BAO

Hence sufficient

Thus C

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masuarezdl: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Thu Jan 29, 2009 9:57 am; Thanked: 1 times

Re: Triangles and Semicircles

by masuarezdl » Sat Jan 31, 2009 12:59 pm

Vikram, the correct answer choice is D, according to the book. But I have no idea why. Because so far, the triangle can be equilatera or isoceles.

I concluded the same as you, but unfortunately it is incorrect.

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ontopofit: Senior | Next Rank: 100 Posts; Posts: 68; Joined: Sun Dec 21, 2008 12:00 am; Thanked: 3 times

by ontopofit » Sat Jan 31, 2009 1:13 pm

this requires a long answer

i will tell in short
statement 1) cod = 60

so, coa = 120
bao + coa + aco = 180
thus bao + aco = 60
aco = cbo
so, bao + cbo =60
cbo = 2bao
hence bao = 20 and suff.

statement 2) aco = 40
aco = cbo
cbo = 2bao
hence bao = 20 and suff.

so answer is D. please just apply the angle sum prop of a triangle and exterior angle prop. of a triangle.

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antoni: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Sat Mar 07, 2009 1:02 am; Location: Russia; Thanked: 2 times

by antoni » Sat Mar 07, 2009 1:09 am

ontopofit wrote:this requires a long answer
i will tell in short
statement 1) cod = 60

so, coa = 120
bao + coa + aco = 180
thus bao + aco = 60
aco = cbo
so, bao + cbo =60
cbo = 2bao
hence bao = 20 and suff.

statement 2) aco = 40
aco = cbo
cbo = 2bao
hence bao = 20 and suff.

so answer is D. please just apply the angle sum prop of a triangle and exterior angle prop. of a triangle.

Hi ontopofit,
I got the answer B but it wasnt the answer. Why I chose B was because I could not understand why:
cbo = 2bao

pls can you explain more? Thanks

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Mar 07, 2009 7:52 am

Here's my visual solution.
Please note that I only address why Statement (1) is sufficient, since there already seems to be agreement/understanding regarding the sufficiency of Statement (2)

Brent Hanneson - Creator of GMATPrepNow.com

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antoni: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Sat Mar 07, 2009 1:02 am; Location: Russia; Thanked: 2 times

by antoni » Sat Mar 07, 2009 8:55 am

Brent Hanneson wrote:Here's my visual solution.
Please note that I only address why Statement (1) is sufficient, since there already seems to be agreement/understanding regarding the sufficiency of Statement (2)

Thank you Brent Hanneson!

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Vemuri: Legendary Member; Posts: 682; Joined: Fri Jan 16, 2009 2:40 amFollowed by: 1 member; Thanked: 32 times

by Vemuri » Sun Mar 08, 2009 1:29 am

Thanks Brent. That was a good explanation.

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trumpet1184: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Tue Jun 23, 2009 8:38 pm

by trumpet1184 » Mon Jul 27, 2009 7:22 pm

This solution makes sense once we establish the fact that AC is a line segment. However, let's back up one step.

How do we know that AC is one continuous line segment (alternatively, how do we know angle ABC = 180 degrees)? I haven't found anything in the question stem that suggests this - we cannot assume this picture to be drawn to scale.

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navalpike: Master | Next Rank: 500 Posts; Posts: 103; Joined: Mon May 04, 2009 11:53 am; Thanked: 6 times

by navalpike » Mon Aug 10, 2009 1:19 pm

Can anyone access the image provided above by Brent. If so, can you please repost? When I try, it literally tells me that the "link is broken"

EDIT: I can access it now.

Last edited by navalpike on Thu Aug 13, 2009 2:45 pm, edited 2 times in total.

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mgshorrGMAT: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Tue Jul 14, 2009 6:15 am; Location: DC; Thanked: 1 times; GMAT Score:710

Help

by mgshorrGMAT » Mon Aug 10, 2009 5:28 pm

Someone please explain why CBO=2BAO...

I thought I was good at geometry until this problem.

Mark

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mgshorrGMAT: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Tue Jul 14, 2009 6:15 am; Location: DC; Thanked: 1 times; GMAT Score:710

Waaaait a second

by mgshorrGMAT » Mon Aug 10, 2009 5:35 pm

I think I figured it out...CBO=2BAO:

1. AB=OC
2. OC=OB (Both are radii)
2.5 OB=AB
3. <BOA=<BAO (Isosceles Triangle)
4. <CBO=2BAO (CBO is the exterior angle of the triangle, equal to the sum of the two non-adjacent angles.)

Let me know if I'm wrong.

Mark

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pkw209: Master | Next Rank: 500 Posts; Posts: 266; Joined: Mon Oct 19, 2009 9:46 pm; Thanked: 8 times; GMAT Score:690

by pkw209 » Mon Feb 01, 2010 5:05 pm

sounds about right. thanks

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tienvunguyen: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Thu Jul 30, 2009 6:58 amFollowed by: 2 members; Location: Madison, WI; Thanked: 1 times; GMAT Score:750

by tienvunguyen » Fri Feb 05, 2010 12:58 pm

Has anyone been able to answer trumpet1184's question? I think this is a valid question. We cannot assume that A,B,C are on a straight line.

Is this one of those GMAT questions where the assumption is so subtle that it is allowed/ignored?

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Feb 05, 2010 1:18 pm

tienvunguyen wrote:Has anyone been able to answer trumpet1184's question? I think this is a valid question. We cannot assume that A,B,C are on a straight line.

Is this one of those GMAT questions where the assumption is so subtle that it is allowed/ignored?

The instructions at the beginning of the Quant section of the GMAT state explicitly that points in a diagram which appear to be in a straight line can be assumed to in fact be in a straight line. So if you see something that looks like a line on your GMAT, it's not a circle or a something - it's definitely a line.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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