BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem
TTP OnDemand GMAT Video Course
Featured Course

Target Test Prep GMAT OnDemand

Scott Woodbury-Stewart’s private virtual classroom — 400 hours of master-class video lessons for the GMAT Focus Edition.

  • 715+ score guarantee — highest in the industry (99th percentile)
  • 52 chapters · 1,500+ lessons · 4,000+ practice questions
  • 400 hours of video · 1,500+ instructor-led HD smartboard lessons
  • 300,000+ students accepted to Harvard, Stanford, Wharton, Booth & Sloan & more
  • 24/7 live support + weekly Zoom office hours with GMAT instructors
  • TTP AI Assist — 24/7 AI-powered virtual tutor for instant help
  • 1,200+ flashcards + AI-powered study assistant & daily calendar
  • OnDemand, LiveTeach & GMAT Bootcamp formats available
  • Also: GRE, SAT Math & Executive Assessment courses
  • MBA Admissions Consulting now available
  • 🏆 2025 EdTech Breakthrough Award: Test Prep Solution Provider of the Year
  • 200,000+ students served
  • 5-day free trial — $0 to start, no auto-billing, cancel anytime
★★★★★ 5.0 (559 reviews)
130-pt guarantee
$0 to start then $127/mo
Start Free 5-Day Trial Read 559 Reviews →

Geometry

Expert replies
Post new topic Post Reply
Previous Topic Next Topic
Senior | Next Rank: 100 Posts
Posts: 46
Joined: Sun Mar 04, 2007 6:27 am
by Tame the CAT » Wed Apr 25, 2007 3:21 pm
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?


20
40
60
80
120

I don't know the answer yet, but whatever I'm getting isn't one of the answer choices. BTW. Triangle ABC is inscribed in a circle.

Thanks!
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Source: — Problem Solving |
Senior | Next Rank: 100 Posts
Posts: 46
Joined: Sun Mar 04, 2007 6:27 am

by Tame the CAT » Wed Apr 25, 2007 3:31 pm
nevermind. I got it. I feel like an idiot!!
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 46
Joined: Sun Mar 04, 2007 6:27 am

by Tame the CAT » Thu Apr 26, 2007 1:03 pm
Here is the official solution. Sorry, I should have posted it. Someone pm-ed me. I should be more considerate for the others studying.

Also look at the attachment
-----------

According to the question, the ratio of the lengths of arc AC to arc AB to arc BC is 6: 2: 1. An inscribed angle cuts off an arc that is twice its measure in degrees. For example, if angle ACB is 30°, minor arc AB is 60° (or 60/360 = 1/6 of the circumference of the circle).

The angles of triangle ABC are all inscribed angles of the circle, so we can deduce the ratio of the angles of triangle ABC from the ratio of the lengths of minor arcs AC, AB and BC. The ratio of the angles will be the same as the ratios of the arcs.

The ratio of angle ABC to angle BCA to angle BAC is also 6: 2: 1. Using the unknown multiplier strategy, whereby a variable can be multiplied by each segment of a ratio to define the actual values described by that ratio, we might say that the interior angles equal 6x: 2x: x.
The sum of the angles in a triangle is 180, so x + 2x + 6x = 180.
9x = 180
x = 20
Angle BCA = 2x or 40

The correct answer is B.
Attachments
triangleincircleQ.jpg
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Legendary Member
Posts: 559
Joined: Tue Mar 27, 2007 1:29 amFollowed by: 2 members
Thanked: 5 times

by Cybermusings » Sat Apr 28, 2007 1:38 am
thanks for the detailed solution!
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 33
Joined: Sat Apr 14, 2007 8:29 am

by ns88 » Sat May 05, 2007 6:34 pm
Bc=x
Ab=2x
Ac=3x

Since arcs, 9x=360

x=40

Since area of triangle is half of circle, x=20

Bca=40
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 122
Joined: Mon Sep 22, 2008 5:11 pm
Thanked: 1 times

by abcdefg » Fri Jul 17, 2009 6:45 pm
The length of minor arc AB is twice the length of minor arc BC and the length of major arc AC is three times the length of minor arc AB.

I'm so confused here. BC = x, AB = 2x, the length of AC = 3(AB) = 6x!. However BC + AB = AC and BC+AB = 3x. How can 3x = 6x?

What am I doing wrong here? It's driving me crazy. Thanks!
ns88 wrote:Bc=x
Ab=2x
Ac=3x

Since arcs, 9x=360

x=40

Since area of triangle is half of circle, x=20

Bca=40
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 122
Joined: Mon Sep 22, 2008 5:11 pm
Thanked: 1 times

by abcdefg » Fri Jul 17, 2009 7:56 pm
I figured it out. I was tricked by the picture into thinking that the major arc went clockwise. Major ARC ac should be going counter clockwise instead.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
MBA Student
Posts: 1194
Joined: Sat Aug 16, 2008 9:42 pmFollowed by: 17 members
Location: Paris, France
Thanked: 71 times
GMAT Score:710

by gmat740 » Fri Jul 17, 2009 9:16 pm
ns88 wrote:Bc=x
Ab=2x
Ac=3x

Since arcs, 9x=360

x=40

Since area of triangle is half of circle, x=20

Bca=40
Good way of solving. In fact I did this in the same way. But just to clarify, it is not the area, but the angle subtended at the center of the circle

Hope this Helps
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Newbie | Next Rank: 10 Posts
Posts: 6
Joined: Mon Apr 13, 2009 8:01 pm

Don't quite understand the major arc concept

by jsk988 » Tue Jul 28, 2009 1:32 am
I'm confused by this problem because when it refers to major arc AC, I thought it was referring to arc ABC. How is it possible that the smaller arc AC is the minor arc? I saw that someone in the forum wrote that it goes counterclockwise, but I'm not sure if that's the rule....

Can someone please explain?
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 338
Joined: Fri Apr 17, 2009 1:49 amFollowed by: 3 members
Thanked: 9 times

by kaulnikhil » Tue Jul 28, 2009 2:19 am
let length of bc be x
length of ab is 2x
and therefore length of ac is 6x
add them up u get circumfrence therefore 9x = 2pi r
x = 2pir/9
we get the length of bc as 2pir/9
similarly we get length of ab as 4 pi /9
and ac as 12pi/9
find the angle subtended at the canter by each of these cords
we get angle by ab as 80 degrees
bc as 40
and ac as 240 degrees
now with center o OA = Oc
so angle ACO = 30
similarly find BCO = 70
subtract aco from BCO to get BCA as 40
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 186
Joined: Sat Sep 02, 2006 7:50 pmFollowed by: 1 member
Thanked: 10 times

by ogbeni » Sun Sep 13, 2009 3:28 am
Folks,

Sorry to resurrect this thread from the dead but I have a question.

How is it that you cannot apply the principle of the "ratio of the sides of a triangle = ratio of the corresponding angles" to right angle triangles such as the 30-60-90 (1-sqrt(3)-2) triangle? The solution to this question uses the ratios of the sides to deduce and calculate the angles by stating that the ratios are 6:2:1, therefore the angles are 120:40:20

Do another set of rules govern right-triangles? Do you see what I mean?

In the 30-60-90 right triangle, the angles 30-60 are in the ratio 1:2 and we know that the ratio of 1:sqrt(3) definitely isn't 1:2

Hope I'm not superconfusing myself here but some insight into this would help me a whole lot.

Thanks
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Site Admin
Posts: 2567
Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members
Thanked: 712 times
GMAT Score:770

by DanaJ » Sun Sep 13, 2009 1:37 pm
Received a PM.

Actually, you are confusing something: the ratio of 6:2:1 does not refer to lengths of sides, it refers to lengths of arcs. That's an entirely different thing, since arcs have much more "in common" with angles than with lengths of sides of triangles.

So basically even if the ratio between the angles of a triangle is 3:2:1, that doesn't mean that this ratio applies to the sides (actually, it won't).
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Post new topic Post Reply
• Page 1 of 1