Author 
Message 
TAGS:

Hide Tags

Director
Joined: 03 Sep 2006
Posts: 871

In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
10 May 2010, 09:37
3
This post received KUDOS
39
This post was BOOKMARKED
Question Stats:
59% (03:03) correct
41% (02:36) wrong based on 734 sessions
HideShow timer Statistics
Attachment:
Circle.jpg [ 20.23 KiB  Viewed 31623 times ]
In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ? A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\)
Official Answer and Stats are available only to registered users. Register/ Login.



Math Expert
Joined: 02 Sep 2009
Posts: 39728

17
This post received KUDOS
Expert's post
18
This post was BOOKMARKED
In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180(<PCO+<QCR)=1807070=40. Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature).
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 04 Apr 2009
Posts: 68
Location: United Kingdom
Schools: Cornell

6
This post received KUDOS
2
This post was BOOKMARKED
I hope this helps. The explanation goes as: We know angle CBA=35 AOB=diameter and hence angle ACB=90 Thus in triangle ACB, angle CAB=55 Now in triangle AOC, OA=OC=Radius and thus angle ACO=55 and angle AOC=70. Similarly we can find angle DOB=70 Thus angle COD=1807070=40 And length of arc calculation is given in the image=2π
Attachments
IMAG0432.jpg [ 344.09 KiB  Viewed 27488 times ]



SVP
Joined: 16 Nov 2010
Posts: 1663
Location: United States (IN)
Concentration: Strategy, Technology

40/180 = x/pi*18/2 x = 2pi Answr  A
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant)
GMAT Club Premium Membership  big benefits and savings



Senior Manager
Joined: 30 Aug 2009
Posts: 287
Location: India
Concentration: General Management

Re: Geometry Problem [#permalink]
Show Tags
08 Jan 2012, 05:46
kotela wrote: Can anyone please help me in solving this problem....... arc OP = arc QR = (70 * 18 pi )/360 as angle OPR = angle QPR (35 degree) angle for arc PQ = 180  (2 arc QP) = 180  140 = 40 length of arc PR = (40 * 18pi)/360 = 2pi



Intern
Joined: 16 Dec 2011
Posts: 49
GMAT Date: 04232012

Re: Geometry Problem [#permalink]
Show Tags
11 Jan 2012, 01:21
kp1811 can u explain this step angle for arc PQ = 180  ( 2arc QP )
how u got that ? plz update



BSchool Forum Moderator
Joined: 23 Jul 2010
Posts: 558
GPA: 3.4
WE: General Management (NonProfit and Government)

Re: Geometry Problem [#permalink]
Show Tags
11 Jan 2012, 02:32
2
This post received KUDOS
kotela wrote: Can anyone please help me in solving this problem....... There are 23 methods, lets see one of them: 1]You are given that OR is diameter=18 (radius =9); therefore arc OR is a semicircle the semicircle is composed of 3 arcarc OP + arc PQ+ arc QR {arc PQ = 180 arc OP + arc QR}2]the lines PQ & OR are parallel , therefore than alternate angles are congruent ( angle PRO = angle=QPR )but these are inscribed angles and we need the central angles to compute the length of arc given by the formula : {central angle (teta) /360 * 2*pi*radius} 3] Central angle of arc QR or arc OP =2 * inscribed angle (2*35= 70)thus arc's QR + OP have central angles 70 +70= 140 4] thus, central angle for arc PQ =180140=40 ; length of arc PQ= 40/360 *2* pi* 9 = 2 piYou may also employ back solving if you are comfortable with that for faster solution
_________________
How to select your BSchool? General Mistakes to Avoid on the GMAT TOP 10 articles on Time Management on the GMAT Thanks = Kudos. Kudos are appreciated
Rules for posting on the verbal forum



Senior Manager
Joined: 30 Aug 2009
Posts: 287
Location: India
Concentration: General Management

Re: Geometry Problem [#permalink]
Show Tags
11 Jan 2012, 02:46
1
This post received KUDOS
pbull78 wrote: kp1811 can u explain this step angle for arc PQ = 180  ( 2arc QP )
how u got that ? plz update we know arc OP = arc QR so we can write either 180 (arc OP + arc QR) or 180  2OP or 180  2QR



Intern
Joined: 01 Mar 2012
Posts: 30
Concentration: Operations, Finance
GPA: 3.3
WE: Engineering (Manufacturing)

In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
10 May 2012, 20:40
Here I'm giving an elaborate solution. Hope it helps. Let A be the center of the circle. We join the mid point of PQ, let's say S with A. Clearly angle ASQ=90 degree. Now angle ROQ= angle PRO= 35 degree As at center any arc subtends double the angle what it subtends at any peripherial point, we can conclude angle RAQ= 70 degree. Hence angle PAQ= 2* angle SAQ= 2*(9070)= 40 degree 180 degree= pi radian 40 degree= 2 pi/9 rad Hence minor arc PQ= 9*2 pi/9 (As radius of the circle=9) Arc PQ= 2 pi Ans(A)



Manager
Joined: 14 Jun 2012
Posts: 65

In the circle above, PQ is parallel to diameter OR, and OR has [#permalink]
Show Tags
30 Jul 2012, 16:03
1
This post received KUDOS
I encountered this problem on one of the tests. It was around my 12th or 13 question. I do admit that this question stumped me and I took some amount of time in solving this question. I started off with the geometric approach but then quickly got tangled in it. I then shifted to approximating the area and using the process of elimination. I am explaining my approach even though it is unconventional and based only on the situation that I was in. assumption pi=3.14 The radius is given as (OR/2)=9. Hence the circumference of the circle is 18pi. Clearly looking at the figure, the length of arc PQ is much smaller than 1/4 the circumference. 18*pi=18*3.14~42+ (read it as something greater than 42) Hence 1/4 of the circumference is ~ 10+ (read as something greater than 10). The length of Arc PQ is much smaller than 1/4 of the circumference as well. Options C and D can be eliminated since C=~10.5 and D=~13.5. These can be clearly eliminated. Similarly Option E : 3pi ~ very close to 10 and hence I eliminated it. After this it was purely on the figure given than I eliminated the other choices. The arc PQ looks to be more like closer to half of the 1/4 i.e. ~5+. I really have no justification as to how I deduced it. Just an observation so that I could guess the answer. Option B [(9pi)/4] comes to be roughly 7 (read as less than 7) Option A (2pi) comes to ~6.5 (read as less than 6.5) which is closer to mu initial estimate of being 5+. Hence I chose A. I am very well aware that on some other day it is very much possible to choose B and to get the question wrong. Also I am not advocating this method in any way Its just that this was an educated guess which I guessed had 50% chance of being right. The above explanations and diagramatic representations have helped me realize my mistake and should such a problem appear again I am dead sure of the mathematical way to solve it.
_________________
My attempt to capture my BSchool Journey in a Blog : tranquilnomadgmat.blogspot.com
There are no shortcuts to any place worth going.



Intern
Joined: 11 Jul 2012
Posts: 46

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
08 Dec 2012, 02:59
Bunuel how do we know that OR is the diameter of the circle?



Math Expert
Joined: 02 Sep 2009
Posts: 39728

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
08 Dec 2012, 04:55



Director
Joined: 29 Nov 2012
Posts: 878

Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180<PCO<QPR=1807070=40.Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature). <PCO AND < QPR ALTERNATE ANGLES? how are those 2 angles equal?
_________________
Click +1 Kudos if my post helped...
Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/
GMAT Prep software What if scenarios http://gmatclub.com/forum/gmatprepsoftwareanalysisandwhatifscenarios146146.html



Math Expert
Joined: 02 Sep 2009
Posts: 39728

fozzzy wrote: Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180<PCO<QPR=1807070=40.Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature). <PCO AND < QPR ALTERNATE ANGLES? how are those 2 angles equal? It should be <PCQ=180(<PCO+<QCR)=1807070=40.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Director
Joined: 29 Nov 2012
Posts: 878

In the circle above, PQ is parallel to diameter OR, and OR has l [#permalink]
Show Tags
21 May 2013, 01:43
< RPQ = 35 degress we have 2 inscribed angles so they are 70 degrees now take one portion of the semicircle and we get 180  70  70 = 40 degrees 40 / 360 = 1/9th Minor arc = Pi D 1/9 * 18 Pi = 2 Pi
_________________
Click +1 Kudos if my post helped...
Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/
GMAT Prep software What if scenarios http://gmatclub.com/forum/gmatprepsoftwareanalysisandwhatifscenarios146146.html



Manager
Joined: 25 Oct 2013
Posts: 169

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
12 Feb 2014, 03:56
1
This post received KUDOS
Bingo! central angle theorem is the rock star! Thanks for the valuable Math book Bunuel! One other thing I missed in this problem is it asks for the minor arc length PQ. NOT length of PQ. Follow up question. Is it possible to find length of the chord PQ with the information given in the question? If so what is the length of PQ?
_________________
Click on Kudos if you liked the post!
Practice makes Perfect.



Current Student
Joined: 17 Oct 2013
Posts: 49
Location: India
Concentration: Strategy, Statistics
WE: Analyst (Computer Software)

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
23 Jun 2014, 00:41
Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180(<PCO+<QCR)=1807070=40. Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature). Bunnel, don't you think that this question asks for a little too much imagination to be on GMAT? Just curious, would you rate this as a 750 range question on the test?



Math Expert
Joined: 02 Sep 2009
Posts: 39728

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
23 Jun 2014, 01:48
Kconfused wrote: Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180(<PCO+<QCR)=1807070=40. Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature). Bunnel, don't you think that this question asks for a little too much imagination to be on GMAT? Just curious, would you rate this as a 750 range question on the test? I'd say it's 650 at most.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 13 Oct 2013
Posts: 135
Concentration: Strategy, Entrepreneurship

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
04 Oct 2014, 12:02
Hi Bunuel, I did not understand how Central angle theorem applied here. Again, according to the central angle theorem above <QCR=2<QPR=70.
Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180(<PCO+<QCR)=1807070=40. Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature).
_________________
 Kindly press +1 Kudos if my post helped you in any way



Math Expert
Joined: 02 Sep 2009
Posts: 39728

Re: In the circle above, PQ is parallel to diameter OR [#permalink]
Show Tags
05 Oct 2014, 02:31
sunita123 wrote: Hi Bunuel, I did not understand how Central angle theorem applied here. Again, according to the central angle theorem above <QCR=2<QPR=70.
Bunuel wrote: In the circle above, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?A. \(2\pi\) B. \(\frac{9\pi}{4}\) C. \(\frac{7\pi}{2}\) D. \(\frac{9\pi}{2}\) E. \(3\pi\) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Let C be the center of the circle. According to the central angle theorem above <PCO=2<PRO=70. As PQ is parallel to OR, then <QPR=<PRO=35. Again, according to the central angle theorem above <QCR=2<QPR=70. <PCQ=180(<PCO+<QCR)=1807070=40. Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\) Answer: A. For more on circle check the circles chapter of Math Book (link in my signature). Red central angle below is twice inscribed blue angle, because they subtend the same arc QR: Attachment:
Untitled.png [ 7.15 KiB  Viewed 19152 times ]
Hope it's clear.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Re: In the circle above, PQ is parallel to diameter OR
[#permalink]
05 Oct 2014, 02:31



Go to page
1 2
Next
[ 33 posts ]



