Last visit was: 02 Sep 2026, 18:40 It is currently 02 Sep 2026, 18:41
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
User avatar
hrish88
User avatar
Joined: 18 Jul 2009
Last visit: 25 Oct 2013
Posts: 14
Own Kudos:
399
 [238]
Given Kudos: 7
Posts: 14
Kudos: 399
 [238]
Right triangle ABC is to be drawn in the xy-plane so that Updated on: 16 Jul 2013, 00:17
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
14
Kudos
Add Kudos
223
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

65% (hard)

Question Stats:

61% (02:17) correct 39% (02:51) wrong based on 1555 sessions

History

Date
Time
Result
Not Attempted Yet
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
ShowHide Answer
Official Answer
C
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 02 Sep 2026
Posts: 113,061
Own Kudos:
838,708
 [83]
Given Kudos: 111,271
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,061
Kudos: 838,708
 [83]
Right triangle ABC is to be drawn in the xy-plane so that 09 Jan 2010, 05:34
32
Kudos
Add Kudos
49
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.2160

D.2916

E.148,824

i ve got it right.but this problem is very time consuming.can anyone suggest shorter method
Show more

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.
avatar
caioguima
avatar
Joined: 02 Jan 2013
Last visit: 07 Jul 2021
Posts: 35
Own Kudos:
175
 [37]
Given Kudos: 2
Schools: Wharton '17 (M) Booth '17 (A) Kellogg '17 (A$) HBS '17 (D) Stanford '17 (D)
GMAT 1: 750 Q51 V40
GPA: 3.2
WE:Consulting (Consulting)
Schools: Wharton '17 (M) Booth '17 (A) Kellogg '17 (A$) HBS '17 (D) Stanford '17 (D)
GMAT 1: 750 Q51 V40
Posts: 35
Kudos: 175
 [37]
Right triangle ABC is to be drawn in the xy-plane so that 25 Jan 2013, 13:03
25
Kudos
Add Kudos
12
Bookmarks
Bookmark this Post
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)
General Discussion
User avatar
hrish88
User avatar
Joined: 18 Jul 2009
Last visit: 25 Oct 2013
Posts: 14
Own Kudos:
399
Given Kudos: 7
Posts: 14
Kudos: 399
Right triangle ABC is to be drawn in the xy-plane so that 09 Jan 2010, 05:50
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.
Show more

OA is C.very nice explanation.you rock man as always.
User avatar
BlitzHN
User avatar
Joined: 07 Aug 2010
Last visit: 26 May 2011
Posts: 33
Own Kudos:
24
Given Kudos: 9
Posts: 33
Kudos: 24
Right triangle ABC is to be drawn in the xy-plane so that 14 Oct 2010, 22:06
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.
Show more

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: 02 Sep 2026
Posts: 113,061
Own Kudos:
838,708
Given Kudos: 111,271
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,061
Kudos: 838,708
Right triangle ABC is to be drawn in the xy-plane so that 15 Oct 2010, 02:58
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....
Show more

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
User avatar
Joined: 25 Jul 2010
Last visit: 21 Aug 2012
Posts: 99
Own Kudos:
107
 [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: 107
 [6]
Right triangle ABC is to be drawn in the xy-plane so that 17 Oct 2010, 20:24
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
User avatar
Joined: 13 Aug 2012
Last visit: 29 Dec 2013
Posts: 316
Own Kudos:
2,125
 [7]
Given Kudos: 11
Concentration: Marketing, Finance
Schools: Full Time MBA (A$)
GPA: 3.23
Schools: Full Time MBA (A$)
Posts: 316
Kudos: 2,125
 [7]
Right triangle ABC is to be drawn in the xy-plane so that 25 Jan 2013, 07:08
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
Show more

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
avatar
Joined: 01 Apr 2013
Last visit: 05 May 2017
Posts: 11
Own Kudos:
12
Given Kudos: 37
Schools: Tepper '16 (S)
Schools: Tepper '16 (S)
Posts: 11
Kudos: 12
Right triangle ABC is to be drawn in the xy-plane so that 18 May 2013, 10:49
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.
Show more

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: 02 Sep 2026
Posts: 113,061
Own Kudos:
838,708
Given Kudos: 111,271
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,061
Kudos: 838,708
Right triangle ABC is to be drawn in the xy-plane so that 19 May 2013, 04:09
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 .
Show more

ANY 3 non-collinear points on a plane form a triangle.
User avatar
cjliu49
User avatar
Joined: 01 Jan 2013
Last visit: 02 May 2014
Posts: 28
Own Kudos:
39
Given Kudos: 13
Location: United States
Concentration: Entrepreneurship, Strategy
Schools: Stanford '16 (D) HBS '16 (D) Wharton '16 (D) Haas '16 (WL)
GMAT 1: 770 Q50 V47
WE:Consulting (Consulting)
Schools: Stanford '16 (D) HBS '16 (D) Wharton '16 (D) Haas '16 (WL)
GMAT 1: 770 Q50 V47
Posts: 28
Kudos: 39
Right triangle ABC is to be drawn in the xy-plane so that 30 May 2013, 21:14
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: 31 Aug 2026
Posts: 7,291
Own Kudos:
17,473
 [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:
My Reviews
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 7,291
Kudos: 17,473
 [2]
Right triangle ABC is to be drawn in the xy-plane so that 19 Sep 2018, 20:58
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
Show more

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 22672 times ]

avatar
ganeshvenugopal
avatar
Joined: 10 Jan 2016
Last visit: 26 Jul 2020
Posts: 5
Given Kudos: 100
Posts: 5
Kudos: 0
Right triangle ABC is to be drawn in the xy-plane so that 28 Sep 2018, 14:32
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.
Show more

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: 31 Aug 2026
Posts: 7,291
Own Kudos:
17,473
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:
My Reviews
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 7,291
Kudos: 17,473
Right triangle ABC is to be drawn in the xy-plane so that 28 Sep 2018, 20:49
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.
Show more

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
User avatar
Joined: 13 Mar 2021
Last visit: 23 Dec 2021
Posts: 303
Own Kudos:
147
Given Kudos: 226
Posts: 303
Kudos: 147
Right triangle ABC is to be drawn in the xy-plane so that 26 May 2021, 06:05
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
User avatar
Joined: 04 May 2020
Last visit: 18 Nov 2025
Posts: 371
Own Kudos:
315
Given Kudos: 308
Status:What goes round comes around, so does Kudos.
Location: India
Schools: Tepper '27 Kelley '27 (D) Kenan-Flagler '27 (D) Jones '27 (WL) McCombs '27 (D)
GPA: 3
WE:Business Development (Retail Banking)
Schools: Tepper '27 Kelley '27 (D) Kenan-Flagler '27 (D) Jones '27 (WL) McCombs '27 (D)
Posts: 371
Kudos: 315
Right triangle ABC is to be drawn in the xy-plane so that 24 Jun 2021, 08:29
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
User avatar
Joined: 18 Apr 2025
Last visit: 02 Sep 2026
Posts: 75
Own Kudos:
49
Given Kudos: 26
Location: India
Schools: IESE '26 (WL)
GMAT Focus 1: 645 Q81 V82 DI83
Products:
GMAT Club Tests Forum Quiz
Schools: IESE '26 (WL)
GMAT Focus 1: 645 Q81 V82 DI83
Posts: 75
Kudos: 49
Right triangle ABC is to be drawn in the xy-plane so that 31 Mar 2026, 09:50
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 501 times ]

Moderator:
Bunuel
Math Expert
113061 posts