GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Nov 2018, 14:02

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

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

Events & Promotions

Events & Promotions in November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
• How to QUICKLY Solve GMAT Questions - GMAT Club Chat

November 20, 2018

November 20, 2018

09:00 AM PST

10:00 AM PST

The reward for signing up with the registration form and attending the chat is: 6 free examPAL quizzes to practice your new skills after the chat.
• The winning strategy for 700+ on the GMAT

November 20, 2018

November 20, 2018

06:00 PM EST

07:00 PM EST

What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.

If a circle passes through points (1, 2), (2, 5), and (5, 4)

Author Message
TAGS:

Hide Tags

Senior Manager
Joined: 06 Aug 2011
Posts: 340
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

11 Feb 2014, 07:11
Thanks again Bunuel
_________________

Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !

Intern
Joined: 05 Feb 2014
Posts: 42

Show Tags

17 Jun 2014, 05:57
lonewolf wrote:
Good method. Simple and quick.

Just for refreshing memories... If two chords of a circle form a right angle degree (for example: AB, BC), then the chord AC must be the diameter of the circle.

Is the above statement always true?

but how do we get to know that the right angle is formed ?
Math Expert
Joined: 02 Sep 2009
Posts: 50670

Show Tags

17 Jun 2014, 06:45
gauravsoni wrote:
lonewolf wrote:
Good method. Simple and quick.

Just for refreshing memories... If two chords of a circle form a right angle degree (for example: AB, BC), then the chord AC must be the diameter of the circle.

Is the above statement always true?

but how do we get to know that the right angle is formed ?

Check here: if-a-circle-passes-through-points-1-2-2-5-and-42105.html#p672194
_________________
Intern
Joined: 05 Feb 2014
Posts: 42

Show Tags

17 Jun 2014, 08:27
Bunuel wrote:
gauravsoni wrote:
lonewolf wrote:
Good method. Simple and quick.

Just for refreshing memories... If two chords of a circle form a right angle degree (for example: AB, BC), then the chord AC must be the diameter of the circle.

Is the above statement always true?

but how do we get to know that the right angle is formed ?

Check here: if-a-circle-passes-through-points-1-2-2-5-and-42105.html#p672194

I understand that if a right triangle is formed then the side will be the diameter. But in the question how do we come to know that a right angle triangle is being formed ?
Math Expert
Joined: 02 Sep 2009
Posts: 50670

Show Tags

18 Jun 2014, 07:51
1
gauravsoni wrote:
Bunuel wrote:
gauravsoni wrote:
but how do we get to know that the right angle is formed ?

Check here: if-a-circle-passes-through-points-1-2-2-5-and-42105.html#p672194

I understand that if a right triangle is formed then the side will be the diameter. But in the question how do we come to know that a right angle triangle is being formed ?

I guess you did not read that post to the end:

The slope of line segment: A(1,2) and B(2,5) is 3 AND the slope of line segment: B(2,5) and C(5,4) is -1/3, the slopes are negative reciprocals hence these line segments are perpendicular to each other. We have right triangle ABC, AC=hypotenuse=Diameter.
_________________
Intern
Joined: 16 May 2015
Posts: 2
Location: India
Concentration: Operations, Finance
GPA: 4
WE: Other (Other)
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

23 Sep 2015, 07:43
Hi Friends,

This Q can b solved under a minute, as the question asks for the diameter, we know that the value must be 2*radius i.e. and among the options mentioned only sqrt(20) can be identified as 2*sqrt(5). Hence option B

Hopes it makes sense.
Manager
Joined: 28 Apr 2016
Posts: 91
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

11 May 2016, 06:39
Since the formula is (x-a)^2 + (y-b)^2 = r^2. How did you take (a-1)^2 + (b-2)^2 = r^2?

Fig wrote:
(B) for me

The equation of circle C in the XY plan is :
(x-a)^2 + (y-b)^2 = r^2

Where :
o (a,b) is the center of the circle
o r is the radius of the circle

Knowing that, u can plug the coordonates of each points to get a, b and r.

(1, 2) on C implies:
(a-1)^2 + (b-2)^2 = r^2
<=> a^2 -2*a + 1 + b^2 - 4*b + 4 = r^2 (1)

(2, 5) on C implies:
(a-2)^2 + (b-5)^2 = r^2
<=> a^2 -4*a + 4 + b^2 - 10*b + 25 = r^2 (2)

(5, 4) on C implies:
(a-5)^2 + (b-4)^2 = r^2
<=> a^2 -10*a + 25 + b^2 -8*b + 16 = r^2 (3)

(1) - (2)
<=> 2*a - 3 + 6*b -21 =0
<=> 2*a + 6*b = 24
<=> a + 3*b = 12 (4)

(1) - (3)
<=> 8*a -24 + 4*b - 12 = 0
<=> 2*a + b = 9 (5)

(5) -2*(4)
<=> -5*b = -15
<=> b = 3

From (1), we find a :
a = 12 - 3*b = 12 - 3*3 = 3

Still from (1):
(a-1)^2 + (b-2)^2 = r^2
<=> (2)^2 + (1)^2 = r^2
<=> r = sqrt(5)

Thus,
D = 2*sqrt(5) = sqrt(20)
Manager
Joined: 05 Sep 2014
Posts: 77
Schools: IIMB
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

23 Jul 2016, 07:49
Bunuel wrote:
gottabwise wrote:
lonewolf: I'd like to know the answer as well. Great question.

I got B. Draw a coord plane, plotted pts, draw right triangle. Used distance formula for each side. Thought about the radius formula for circumscribed circles. Nixed idea. Noticed isoceles right triangle and realized sqrt 20 was diameter b/c it's the hypotenuse. Ended there. Glad I tried the problem.

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.
Attachment:
Math_Tri_inscribed.png

So: If two chords of a circle form a right angle degree (for example: AB, BC), then the chord AC must be the diameter of the circle. TRUE.

As for the question: the slope of line segment: A(1,2) and B(2,5) is 3 AND the slope of line segment: B(2,5) and C(5,4) is -1/3, the slopes are negative reciprocals hence these line segments are perpendicular to each other. We have right triangle ABC, AC=hypotenuse=Diameter.

Also one tip: any three points, which are not collinear, define the unique circle on XY-plane. For more please see the Triangles and Circles chapters of Math Book in my signature.

Hope it's clear.

Hi Bunnuel,

Thanks a lot for your explanation, really helped me. However just wanted to understand how do we calculate the diameter if its not the right angle triangle in a circle. Apologies if this sounds too basic. I made a graph of the same, unable to analyze what could be diameter in such question.

Regards
Megha
Senior Manager
Status: You have to have the darkness for the dawn to come
Joined: 09 Nov 2012
Posts: 295
Daboo: Sonu
GMAT 1: 590 Q49 V20
GMAT 2: 730 Q50 V38
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

19 Mar 2017, 20:26
bz9 wrote:
If a circle passes through points $$(1, 2)$$, $$(2, 5)$$, and $$(5, 4)$$, what is the diameter of the circle?

A. $$\sqrt{18}$$
B. $$\sqrt{20}$$
C. $$\sqrt{22}$$
D. $$\sqrt{26}$$
E. $$\sqrt{30}$$

Let us consider the origin is at (x,y)
so from distance formula
(x-1)^2+ (y-2)^2 =r^2 ..................................1
(x-2)^2+(y-5)^2= r^2 ..................................2
(x-5)^2+(x-4)^2 =r^2 ..................................3
on solving 1 and 2 we get x+3y =12 ..................4
on solving 1 and 3 we get 2x+ 6y=9 ..................5
now solve 4 and 5
(x,y)= (3,3) and is the center
Sqrt((3-1)^2+(3-2)^2)= sqrt( 4+ 1)= sqrt(5)= radius
Diamiter = 2*radius = 2* sqrt(5) = sqrt(20)
hence B
_________________

You have to have the darkness for the dawn to come.

Give Kudos if you like my post

Intern
Joined: 24 Oct 2014
Posts: 16
Location: Spain
Schools: LBS '21 (II)
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

15 Jan 2018, 02:32
I have a question:

It is a PS question, so the stem has to be enough to answer the question.
If ABC wasn't a right triangle, it wouldn't be possible.

Therefore, I believe checking whether there is or not a right angle is not necessary and we should jump straight to calculating the distance between (1,2) and (5,4) because it has to be the hypotenuse (ergo the diameter).

Do you agree?

Thank you.
Senior Manager
Joined: 31 Jul 2017
Posts: 494
Location: Malaysia
GMAT 1: 700 Q50 V33
GPA: 3.95
WE: Consulting (Energy and Utilities)
If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

25 Jan 2018, 12:18
bz9 wrote:
If a circle passes through points $$(1, 2)$$, $$(2, 5)$$, and $$(5, 4)$$, what is the diameter of the circle?

A. $$\sqrt{18}$$
B. $$\sqrt{20}$$
C. $$\sqrt{22}$$
D. $$\sqrt{26}$$
E. $$\sqrt{30}$$

$$1st Method:$$

The equation of circle with $$(h,k)$$ as centre is given by, $$(x-h)^2 + (y-k)^2 = r^2$$
As per the given equation,
$$(1-h)^2+(2-k)^2 = r^2 ---------1$$
$$(2-h)^2 + (5-k)^2 = r^2---------2$$
$$(5-h)^3 + (4-k)^2 = r^2---------3$$

Solving above 3 equations we get, $$(h,k) = (3,3)$$. Therefore, $$D = \sqrt{20}.$$

$$2nd Method:$$

The Points A, B & C form a right angle traingle. Hnece, the diameter will be equal to the distance between the Points forming the Hypotenous.
_________________

If my Post helps you in Gaining Knowledge, Help me with KUDOS.. !!

Intern
Joined: 01 Feb 2018
Posts: 18
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

28 Mar 2018, 10:17
How do we exactly know that the line combining these two points go through the center? What if (in another question) it does not go through the center?
Intern
Joined: 02 Feb 2018
Posts: 23
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4)  [#permalink]

Show Tags

11 Nov 2018, 08:35
Fernandocma wrote:
I have a question:

It is a PS question, so the stem has to be enough to answer the question.
If ABC wasn't a right triangle, it wouldn't be possible.

Therefore, I believe checking whether there is or not a right angle is not necessary and we should jump straight to calculating the distance between (1,2) and (5,4) because it has to be the hypotenuse (ergo the diameter).

Do you agree?

Thank you.

Agree, that's also what I thought
Re: If a circle passes through points (1, 2), (2, 5), and (5, 4) &nbs [#permalink] 11 Nov 2018, 08:35

Go to page   Previous    1   2   [ 33 posts ]

Display posts from previous: Sort by