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Right triangle ABC is to be drawn in the xy-plane so that [#permalink]
 09 Jan 2010, 04:50
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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
Last edited by Bunuel on 02 Apr 2012, 18:29, edited 2 times in total.
Edited the question and added the OA
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hrish88 wrote: 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.
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Bunuel wrote: hrish88 wrote: 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. Good one. +1 for it. Hope I didn't mess it up. OA is C.very nice explanation.you rock man as always.
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Bunuel wrote: hrish88 wrote: 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....
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BlitzHN wrote: Bunuel wrote: hrish88 wrote: 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.... 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=dimensionsHope it helps.
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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.
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Re: Right triangle ABC is to be drawn in the xy-plane so that [#permalink]
25 Jan 2013, 07:08
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hrish88 wrote: 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 = 2,160Answer: C
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Re: Right triangle ABC is to be drawn in the xy-plane so that [#permalink]
25 Jan 2013, 13:03
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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)
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Bunuel wrote: hrish88 wrote: 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. 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 .
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venkat18290 wrote: Bunuel wrote: hrish88 wrote: 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. 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.
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PLEASE READ AND FOLLOW: 11 Rules for Posting!!!
RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory
COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!
DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!
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