Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 18 Jul 2009
Posts: 26

Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
Updated on: 16 Jul 2013, 00:17
Question Stats:
59% (02:19) correct 41% (02:52) wrong based on 891 sessions
HideShow timer Statistics
Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions? A. 54 B. 432 C. 2,160 D. 2,916 E. 148,824
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by hrish88 on 09 Jan 2010, 04:50.
Last edited by Bunuel on 16 Jul 2013, 00:17, edited 3 times in total.
Edited the question and added the OA




Math Expert
Joined: 02 Sep 2009
Posts: 60480

Re: tough problem
[#permalink]
Show Tags
09 Jan 2010, 05:34
hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C.
_________________




Manager
Joined: 02 Jan 2013
Posts: 54
GPA: 3.2
WE: Consulting (Consulting)

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
25 Jan 2013, 13:03
Slightly different way of thinking:
On the 9x6 grid of possibilities, I can imagine a bunch of rectangles (with sides parallel to x and y axes). Each of these rectangles contains 4 triangles that fit the description of the question stem.
therefore:
Answer = ( # of Rectangles I can make on the grid) x 4
To create the rectangle, I need to pick 2 points on the x direction, and 2 points on the y direction. Therefore:
Answer = C(9,2) * C(6,2) * 4 = 36 * 15 * 4 = 2160 (OPTION C)




Intern
Joined: 18 Jul 2009
Posts: 26

Re: tough problem
[#permalink]
Show Tags
09 Jan 2010, 05:50
Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the square with dimensions 9*6(9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. OA is C.very nice explanation.you rock man as always.



Manager
Joined: 07 Aug 2010
Posts: 50

Re: tough problem
[#permalink]
Show Tags
14 Oct 2010, 22:06
Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Good one. +1 for it. Hope I didn't mess it up. so what about the triangles that look like the mirror images of the ones above?  think, switching the coords of A and C along x axis and switching A and B along y axis....



Math Expert
Joined: 02 Sep 2009
Posts: 60480

Re: tough problem
[#permalink]
Show Tags
15 Oct 2010, 02:58
BlitzHN wrote: Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. so what about the triangles that look like the mirror images of the ones above?  think, switching the coords of A and C along x axis and switching A and B along y axis.... Above solution counts all position: AC and CA; A B and B A; For example point C with 8C1 can be placed to the right as well to the left of A and point B with 5C1 can be placed below as well as above of A. So all cases are covered. More here: arithmeticogquestion88380.html?hilit=dimensionsHope it helps.
_________________



Manager
Status: ISB, Hyderabad
Joined: 25 Jul 2010
Posts: 120
WE 1: 4 years Software Product Development
WE 2: 3 years ERP Consulting

Re: tough problem
[#permalink]
Show Tags
17 Oct 2010, 20:24
C.
I am not sure if this approach is correct. I used Elimination. There can be only 5 possible values of C if we fix A. So the number of triangles possible has to be multiple of 5. The only answer that satisfies the criterion is C.



Senior Manager
Joined: 13 Aug 2012
Posts: 395
Concentration: Marketing, Finance
GPA: 3.23

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
25 Jan 2013, 07:08
hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A. 54 B. 432 C. 2,160 D. 2,916 E. 148,824 First, get the integer points available for xaxis: 2  (6) + 1 = 9 Second, get the interger points available for yaxis: 94+1 = 6 How many ways to select the location of line AB in the xaxis? 9 How many ways to select the location of point C in the xaxis? 8 (Note: we cannot select the location of line AB) How many ways to select the location of the base? 2 (Is it BC or AB?) How many ways to position line AB parallel to y axis? 6!/2!4! = 15 Multiple all that:\(9*8*2*15 = 2,160\) Answer: C



Intern
Joined: 01 Apr 2013
Posts: 16

Re: tough problem
[#permalink]
Show Tags
18 May 2013, 10:49
Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Hi Bunuel , That was a fantastic solution , but i have a small doubt . How do we ensure that by selecting points in this way the properties of a triangle are satisfied always . Could there be some points through which we cannot even form a triangle leave alone right angled triangle. I hope i am clear in my question .



Math Expert
Joined: 02 Sep 2009
Posts: 60480

Re: tough problem
[#permalink]
Show Tags
19 May 2013, 04:09
venkat18290 wrote: Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Hi Bunuel , That was a fantastic solution , but i have a small doubt . How do we ensure that by selecting points in this way the properties of a triangle are satisfied always . Could there be some points through which we cannot even form a triangle leave alone right angled triangle. I hope i am clear in my question . ANY 3 noncollinear points on a plane form a triangle.
_________________



Intern
Joined: 01 Jan 2013
Posts: 29
Location: United States
Concentration: Entrepreneurship, Strategy
WE: Consulting (Consulting)

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
30 May 2013, 21:14
Bunuel... you're a freaking genius. Get a job with NASA already.



Senior Manager
Joined: 03 Apr 2013
Posts: 262
Location: India
Concentration: Marketing, Finance
GMAT 1: 740 Q50 V41
GPA: 3

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
13 Nov 2013, 08:38
Another way of looking at the problem. According to the given constraints, the coordinates have to be chosen this way : A(a,b) B(a,c) C(d,b) where a,b,c and d are arbitrary integers. If you check this satisfies the constraint that AB must be parallel to the Yaxis. Drawing the triangle and rotating it will give you a rectangle whose sides will measure length= bc and breadth= ad. This rectangle's area will be = bc X ad Now after having realized this, you just have to choose values from the given ranges such that the area is always nonzero, and this can be done in the following way, !. selecting a and d from the range [6,2] which has 9 elements, derived as > 2  (6) +1 = 9. 9C2 X 2 (2 because both a>d and d>a are permissible). 2. selecting b and c similarly 6C2 X 2. 3. Multiplying the two terms : 9C2 X 6C2 X 2 X 2 = 2160. Kudos if you liked it. Do have a look at this approach Bunuel



Intern
Joined: 22 Jun 2013
Posts: 11

Re: tough problem
[#permalink]
Show Tags
13 Nov 2013, 09:38
Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Kudos Bunuel. Nice explanation.



BSchool Forum Moderator
Joined: 23 May 2018
Posts: 683
Location: Pakistan

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
26 Jun 2018, 01:40
Making a rough diagram for the axis helps.



CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2981
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
19 Sep 2018, 20:58
hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A. 54 B. 432 C. 2,160 D. 2,916 E. 148,824 Please check solution as attached. Answer:Option C
Attachments
File comment: www.GMATinsight.com
Screen Shot 20180920 at 9.26.09 AM.png [ 659.51 KiB  Viewed 4942 times ]
_________________
Prosper!!!GMATinsightBhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhihttp://www.GMATinsight.com/testimonials.htmlACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Intern
Joined: 10 Jan 2016
Posts: 5

Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
28 Sep 2018, 14:32
Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Hi @bunnel , I have a doubt. If AB is parallel to Yaxis, how can we count X=0 as a possibility for vertex A? If X=0, when vertex A lies of the Yaxis and therefore, AB can't be parallel to Yaxis.. With this in mind, I got 8*6 possibilities for vertex A. please help.



CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2981
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
28 Sep 2018, 20:49
ganeshvenugopal wrote: Bunuel wrote: hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A.54
B.432
C.2160
D.2916
E.148,824
i ve got it right.but this problem is very time consuming.can anyone suggest shorter method We have the rectangle with dimensions 9*6 (9 horizontal dots and 6 vertical). AB is parallel to yaxis and AC is parallel to xaxis. Choose the (x,y) coordinates for vertex A: 9C1*6C1; Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (91=8 as 1 horizontal dot is already occupied by A); Choose the y coordinate for vertex B (as x coordinate is fixed by A): 5C1, (61=5 as 1 vertical dot is already occupied by A). 9C1*6C*8C1*5C1=2160. Answer: C. Hi @bunnel , I have a doubt. If AB is parallel to Yaxis, how can we count X=0 as a possibility for vertex A? If X=0, when vertex A lies of the Yaxis and therefore, AB can't be parallel to Yaxis.. With this in mind, I got 8*6 possibilities for vertex A. please help. Here comes some cotradiction. One definition says that parallel lines are lines that never intersect but they lack mentioning the fact that the lines should lie in one plane for it to happen. Here the line is parallel to Y axis and YAxis is a direction while origin is just a point of reference from where the Y direction may be referenced so I believe that X=0 may be taken for the line which is parallel to YAxis.
_________________
Prosper!!!GMATinsightBhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhihttp://www.GMATinsight.com/testimonials.htmlACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Intern
Joined: 09 Aug 2018
Posts: 11

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
18 Mar 2019, 12:00
BunuelThank you for your solution. I have a question: How can we ensure that the sum of any 2 sides of the triangle is greater than the third side and that the length is greater than the difference between the lengths of the other 2 sides? Thank you!



Manager
Joined: 19 Jan 2018
Posts: 85

Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
Show Tags
11 Dec 2019, 20:34
hrish88 wrote: Right triangle ABC is to be drawn in the xyplane so that the right angle is at A and AB is parallel to the yaxis. If the x and ycoordinates of A, B, and C are to be integers that are consistent with the inequalities 6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A. 54 B. 432 C. 2,160 D. 2,916 E. 148,824 A very simple solution that you can solve in less than 30 seconds. If you look at the answer choices and the ranges for the Ycoordinates 4 ≤ y ≤ 9 This means that there are 6 integers, and each side of a triangle has to take 2 points. C(6,2) = 15 Because 15 is a multiple of 5, the answer choice has to end with a 0 or 5. Only C does so




Re: Right triangle ABC is to be drawn in the xyplane so that
[#permalink]
11 Dec 2019, 20:34






