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A right triangle ABC has to be constructed in the xy-plane

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A right triangle ABC has to be constructed in the xy-plane [#permalink] New post 08 Sep 2010, 12:21
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A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordiantes of A, B and C are to satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. The numbers of different triangles that can be constructed with these properties are?

A. 63
B. 336
C. 567
D. 3024
E. 5040
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Re: Geometry and Combinations [#permalink] New post 08 Sep 2010, 12:36
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zisis wrote:
A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordiantes of A, B and C are to satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. The numbers of different triangles that can be constructed with these properties are?

A63
B336
C567
D3024
E5040


Question should specify that x- and y-coordinates of A, B, and C are to be integers.

We have the rectangle with dimensions 9*7 (9 horizontal dots and 7 vertical). AC is parallel to y-axis and AB is parallel to x-axis.

Choose the (x,y) coordinates for vertex A (right angle): 9C1*7C1;
Choose the x coordinate for vertex B (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 C (as x coordinate is fixed by A): 6C1, (7-1=6 as 1 vertical dot is already occupied by A).

9C1*7C1*8C1*6C1=3024.

Answer: D.
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Re: Geometry and Combinations [#permalink] New post 13 Oct 2010, 22:36
zisis wrote:
A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordiantes of A, B and C are to satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. The numbers of different triangles that can be constructed with these properties are?

A63
B336
C567
D3024
E5040


Note: A and C share the same x coordinate. A and B share the same y coordinate.

The x coordinates have 9 options (x = -1,0,1,2,3,4,5,6,7) pick 2: 1 for point B, and 1 for point A and C. 9P2
The y coordinates have 7 options (y= 1,2,3,4,5,6,7) pick 2: 1 for point A and B, and 1 for point C. 7P2
Number of triangles w/ A as the right angle = 9P2 * 7P2 = 3024
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Re: Geometry and Combinations [#permalink] New post 14 Oct 2010, 00:59
Good question.+1 KUDO
Re: Geometry and Combinations   [#permalink] 14 Oct 2010, 00:59
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