Last visit was: 22 Apr 2026, 10:51 It is currently 22 Apr 2026, 10:51
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
User avatar
hrish88
Joined: 18 Jul 2009
Last visit: 25 Oct 2013
Posts: 14
Own Kudos:
396
 [236]
Given Kudos: 7
Posts: 14
Kudos: 396
 [236]
14
Kudos
Add Kudos
221
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,748
Own Kudos:
Given Kudos: 105,821
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,748
Kudos: 810,639
 [83]
32
Kudos
Add Kudos
49
Bookmarks
Bookmark this Post
avatar
caioguima
Joined: 02 Jan 2013
Last visit: 07 Jul 2021
Posts: 35
Own Kudos:
175
 [37]
Given Kudos: 2
GMAT 1: 750 Q51 V40
GPA: 3.2
WE:Consulting (Consulting)
25
Kudos
Add Kudos
12
Bookmarks
Bookmark this Post
General Discussion
User avatar
hrish88
Joined: 18 Jul 2009
Last visit: 25 Oct 2013
Posts: 14
Own Kudos:
Given Kudos: 7
Posts: 14
Kudos: 396
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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.
User avatar
BlitzHN
Joined: 07 Aug 2010
Last visit: 26 May 2011
Posts: 33
Own Kudos:
Given Kudos: 9
Posts: 33
Kudos: 24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 co-ords of A and C along x axis and switching A and B along y axis....
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,748
Own Kudos:
Given Kudos: 105,821
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,748
Kudos: 810,639
Kudos
Add Kudos
Bookmarks
Bookmark this Post
BlitzHN
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 co-ords 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: arithmetic-og-question-88380.html?hilit=dimensions

Hope it helps.
User avatar
arundas
Joined: 25 Jul 2010
Last visit: 21 Aug 2012
Posts: 99
Own Kudos:
106
 [6]
Given Kudos: 15
Status:ISB, Hyderabad
Concentration: Strategy
WE 1: 4 years Software Product Development
WE 2: 3 years ERP Consulting
Posts: 99
Kudos: 106
 [6]
5
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
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.
User avatar
mbaiseasy
Joined: 13 Aug 2012
Last visit: 29 Dec 2013
Posts: 316
Own Kudos:
2,094
 [7]
Given Kudos: 11
Concentration: Marketing, Finance
GPA: 3.23
Posts: 316
Kudos: 2,094
 [7]
4
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 x-axis: 2 - (-6) + 1 = 9
Second, get the interger points available for y-axis: 9-4+1 = 6

How many ways to select the location of line AB in the x-axis? 9
How many ways to select the location of point C in the x-axis? 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
avatar
venkat18290
Joined: 01 Apr 2013
Last visit: 05 May 2017
Posts: 11
Own Kudos:
Given Kudos: 37
Posts: 11
Kudos: 12
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 .
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,748
Own Kudos:
Given Kudos: 105,821
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,748
Kudos: 810,639
Kudos
Add Kudos
Bookmarks
Bookmark this Post
venkat18290
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 non-collinear points on a plane form a triangle.
User avatar
cjliu49
Joined: 01 Jan 2013
Last visit: 02 May 2014
Posts: 28
Own Kudos:
Given Kudos: 13
Location: United States
Concentration: Entrepreneurship, Strategy
GMAT 1: 770 Q50 V47
WE:Consulting (Consulting)
GMAT 1: 770 Q50 V47
Posts: 28
Kudos: 39
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel... you're a freaking genius. Get a job with NASA already.
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 22 Apr 2026
Posts: 6,976
Own Kudos:
16,899
 [2]
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,976
Kudos: 16,899
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 2018-09-20 at 9.26.09 AM.png
Screen Shot 2018-09-20 at 9.26.09 AM.png [ 659.51 KiB | Viewed 22240 times ]

avatar
ganeshvenugopal
Joined: 10 Jan 2016
Last visit: 26 Jul 2020
Posts: 5
Given Kudos: 100
Posts: 5
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 Y-axis, how can we count X=0 as a possibility for vertex A?
If X=0, when vertex A lies of the Y-axis and therefore, AB can't be parallel to Y-axis..
With this in mind, I got 8*6 possibilities for vertex A.

please help.
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 22 Apr 2026
Posts: 6,976
Own Kudos:
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,976
Kudos: 16,899
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ganeshvenugopal
Bunuel
hrish88
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates 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 y-axis and AC is parallel to x-axis.

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, (9-1=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, (6-1=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 Y-axis, how can we count X=0 as a possibility for vertex A?
If X=0, when vertex A lies of the Y-axis and therefore, AB can't be parallel to Y-axis..
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 Y-Axis 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 Y-Axis.
User avatar
Bambi2021
Joined: 13 Mar 2021
Last visit: 23 Dec 2021
Posts: 306
Own Kudos:
Given Kudos: 226
Posts: 306
Kudos: 142
Kudos
Add Kudos
Bookmarks
Bookmark this Post
If I pick a point for the right angle (A,A), I can pick this point in 9*6 ways. The other two points that forms the triangle must have one point at (A,y) and one point at (x,A). The total number of ways to pick the points becomes 9*6*8*5 = 2160 triangles.
User avatar
Thelegend2631
Joined: 04 May 2020
Last visit: 18 Nov 2025
Posts: 371
Own Kudos:
Given Kudos: 308
Status:What goes round comes around, so does Kudos.
Location: India
GPA: 3
WE:Business Development (Retail Banking)
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Once you know it forms a 6*9 Rectangle - IT becomes easy.

Since AB - Parallel to Y Axis -

Select two points ( AB ) from 6 vertical points : 6C2
- Multiply this by 2 Since there can be right angle above or below
- Multiply this by 9 Since AB Can be anywhere on 9 vertical bars
- Multiple this with 8 since we have 8 choices for Point C

6c2*2*8*9 = 2160
.
User avatar
yashsharma1
Joined: 18 Apr 2025
Last visit: 22 Apr 2026
Posts: 13
Given Kudos: 3
Location: India
GMAT Focus 1: 645 Q81 V82 DI83
GMAT Focus 1: 645 Q81 V82 DI83
Posts: 13
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
This is how I went about solving this.
In the triangle, there will be 3 points A, B & C. Each point will have (x,y) coordinates. Now let's fix point B. There are total 54 (x,y) combinations possible for point B (9 values of x * 6 values of y as given in question)

Let B be situated at (-6,4). Now A is vertically in same line as B (because AB is parallel to y-axis) and hence its x coordinate is fixed as -6. Now for y coordinate, A can have any value except 4 (because otherwise point B will lie on point A and no triangle will be formed) from the possible values allowed as per question. That leaves 5 possible y coordinates for point A.

Similarly, point C is horizontal to A and hence y coordinate for C is fixed at 4. For x coordinate, B can have any value other than -6 and that allows for 8 values of x coordinate for point C.

So, for each combination of (x,y) for point B, there are 40 (5 possibilities of y coordinate for point A and 8 possibilities of x coordinate for point C) different combinations of positioning point B & C. And because point B itself has 54 combinations, final answer is 54*40 = 2160

This is what Bunuel has explained crisply with combination formulas
Attachments

Triangle ABC.jpeg
Triangle ABC.jpeg [ 177.68 KiB | Viewed 76 times ]

Moderators:
Math Expert
109748 posts
Tuck School Moderator
853 posts