Last visit was: 25 Jul 2024, 08:26 It is currently 25 Jul 2024, 08:26
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Show Tags
Hide Tags
User avatar
Manager
Manager
Joined: 18 Jun 2010
Posts: 230
Own Kudos [?]: 677 [133]
Given Kudos: 194
Schools:Chicago Booth Class of 2013
Send PM
Most Helpful Reply
Math Expert
Joined: 02 Sep 2009
Posts: 94614
Own Kudos [?]: 643806 [35]
Given Kudos: 86753
Send PM
General Discussion
User avatar
Director
Director
Joined: 18 Jul 2010
Status:Apply - Last Chance
Affiliations: IIT, Purdue, PhD, TauBetaPi
Posts: 536
Own Kudos [?]: 362 [2]
Given Kudos: 15
Concentration: $ Finance $
Schools:Wharton, Sloan, Chicago, Haas
 Q50  V37
GPA: 4.0
WE 1: 8 years in Oil&Gas
Send PM
User avatar
Intern
Intern
Joined: 14 Apr 2010
Posts: 19
Own Kudos [?]: 73 [0]
Given Kudos: 1
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
Let us assume A to be (1,1).Then, distance from origin = sqrt 2 = Diagonal of square = Diameter of circle.

Area of square = 1 sq units and
Area of circle = Pi r^2 = Pi (sqrt 2 / 2)^2 = Pi / 2

Area of 4 bulging sides = (Pi/2) - 1 = (Pi - 2) / 2 and Area of 2 bulging side = (Pi - 2) / 4.

Therefore, required fraction = Area of 2 bulging side / Area of circle = [(Pi - 2) / 4] / (Pi/2) = Pi -2 / 2Pi.

But answer posted by Bunuel is : Pi + 2 / 2Pi and I take his explainations as absolute. Since I got a different answer....please explain where I am wrong.
User avatar
Manager
Manager
Joined: 10 Nov 2010
Posts: 129
Own Kudos [?]: 2678 [0]
Given Kudos: 22
Location: India
Concentration: Strategy, Operations
GMAT 1: 520 Q42 V19
GMAT 2: 540 Q44 V21
WE:Information Technology (Computer Software)
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong
Math Expert
Joined: 02 Sep 2009
Posts: 94614
Own Kudos [?]: 643806 [2]
Given Kudos: 86753
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
2
Kudos
Expert Reply
GMATD11 wrote:
Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong


Solution: in-a-rectangular-coordinate-system-point-a-has-coordinates-99510.html#p767285
OA: D.
User avatar
Manager
Manager
Joined: 13 Feb 2010
Status:Prevent and prepare. Not repent and repair!!
Posts: 145
Own Kudos [?]: 428 [1]
Given Kudos: 282
Location: India
Concentration: Technology, General Management
GPA: 3.75
WE:Sales (Telecommunications)
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
1
Kudos
Hi, Cant we do this by calculating the area of the sector and then the area of the circle??? Pls help me understand!
User avatar
Manager
Manager
Joined: 15 Jun 2010
Posts: 242
Own Kudos [?]: 1189 [0]
Given Kudos: 50
Concentration: Marketing
 Q47  V26 GMAT 2: 540  Q45  V19 GMAT 3: 580  Q48  V23
GPA: 3.2
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
es we can do it by taking any value for (d,d) and calculating the area deducting the sector.
Take (d,d) as 2,2. the dia meter becomes 2 root 2 and radius root2
so area of circle = 2pie
area of 2 sectors= 2*(90/360)*2 pie = pie
But have to add another half of square = 1/2 * 4 = 2
So area in 1st quadrant = 2pie-pie+2= pie+2
and area of circle = 2pie
So ratio= (2+pie)/2pie
avatar
Intern
Intern
Joined: 22 Oct 2012
Status:K... M. G...
Posts: 23
Own Kudos [?]: 19 [0]
Given Kudos: 118
Concentration: General Management, Leadership
GMAT Date: 08-27-2013
GPA: 3.8
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is \(C=\pi{r^2}=\pi\);
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

Ratio of the are of this region to area of a circle is \(\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}\).

Answer: D.

Hope it's clear.



Hi,

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..
User avatar
Senior Manager
Senior Manager
Joined: 27 Jun 2012
Posts: 323
Own Kudos [?]: 2496 [1]
Given Kudos: 185
Concentration: Strategy, Finance
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
1
Bookmarks
FTG wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is \(C=\pi{r^2}=\pi\);
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

Ratio of the are of this region to area of a circle is \(\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}\).

Answer: D.

Hope it's clear.



Hi,

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..


The question asks for "fraction" or "proportion" of the circle. (e.g. say 1/2 or 3/5th of circle lies in the first quadrant)
Hence you need to take the ratio of Area (in 1st quadrant) to Total area to find the fraction.
User avatar
Intern
Intern
Joined: 03 Jan 2013
Posts: 15
Own Kudos [?]: [0]
Given Kudos: 50
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is \(C=\pi{r^2}=\pi\);
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

Ratio of the are of this region to area of a circle is \(\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}\).

Answer: D.

Hope it's clear.


Quote:
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);


is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?
Math Expert
Joined: 02 Sep 2009
Posts: 94614
Own Kudos [?]: 643806 [1]
Given Kudos: 86753
Send PM
Re: What fraction of the area of circle C lies within ... [#permalink]
1
Kudos
Expert Reply
pharm wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is \(C=\pi{r^2}=\pi\);
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);

Area of a circle without the red parts is \(C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}\);

Ratio of the are of this region to area of a circle is \(\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}\).

Answer: D.

Hope it's clear.


Quote:
Area of a square is half of the product of diagonals, as diagonal equals to \(2r=2\), then \(S=\frac{2^2}{2}=2\);


is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?


If I understand correctly you are asking about the area of a square: \(area_{square}=side^2=\frac{diagonal^2}{2}\). This is a general formula for the area of any square.

Hope it helps.
User avatar
Intern
Intern
Joined: 03 Jan 2013
Posts: 15
Own Kudos [?]: [0]
Given Kudos: 50
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
Yea, thank you cleared things up
Math Expert
Joined: 02 Sep 2009
Posts: 94614
Own Kudos [?]: 643806 [1]
Given Kudos: 86753
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
1
Kudos
Expert Reply
pharm wrote:
Yea, thank you cleared things up


Forgot to mention that the area of a rhombus is also equals to half of the product of diagonals: \(area_{rhombus}=\frac{d_1*d_2}{2}\), where \(d_1\) and \(d_2\) are the lengths of the diagonals (or \(bh\), where \(b\) is the length of the base and \(h\) is the altitude).
User avatar
Manager
Manager
Joined: 05 Sep 2014
Posts: 52
Own Kudos [?]: 11 [0]
Given Kudos: 254
Schools: IIMB
Send PM
In a rectangular coordinate system, point A has coordinates [#permalink]
why we are considering square for this question, its nowhere specified in the question. I do not get this at all, please help.

Regards
Megha
avatar
Intern
Intern
Joined: 29 Mar 2015
Posts: 8
Own Kudos [?]: -12 [0]
Given Kudos: 15
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
A quick look at the answer choices tells you that the ratio does not depend on the value of "d". Hence you can choose "smart numbers" for d - For example 2. Which makes the radius of the circle sqrt(2). The all we need to do is add the area of the semi circle with the area of the triangle to get the area in the first quadrant (refer the diagram by Bunuel).

Solve and the answer is D.
avatar
Intern
Intern
Joined: 29 Mar 2015
Posts: 8
Own Kudos [?]: -12 [0]
Given Kudos: 15
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
megha_2709 wrote:
why we are considering square for this question, its nowhere specified in the question. I do not get this at all, please help.

Regards
Megha


If you drop perpendiculars to x & y axes from (d,d) you will get (d,0) and (0,d). These two points are "d" units away from the origin as well as from (d,d). Knowing that all sides are of the same length and that the 3 angles are 90 degrees, (we drew perpendiculars to the axes and the axes themselves are at a () degree angle) we can conclude that the figure is a square.

Hope it helps.
Manager
Manager
Joined: 12 Jan 2019
Posts: 91
Own Kudos [?]: 58 [1]
Given Kudos: 211
Location: India
Concentration: Finance, Technology
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
1
Bookmarks
rajathpanta wrote:
Hi, Cant we do this by calculating the area of the sector and then the area of the circle??? Pls help me understand!

Hey we cannot do this because we can use sectors only when the angle formed by the arc is at the centre of the circle.
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 34090
Own Kudos [?]: 853 [0]
Given Kudos: 0
Send PM
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
Re: In a rectangular coordinate system, point A has coordinates [#permalink]
Moderator:
Math Expert
94614 posts