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Triangle ABC will be constructed in a xy-plane according to

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Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 14 May 2012, 15:32
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Triangle ABC will be constructed in a xy-plane according to the following conditions: Angle ABC is 90 degrees, and AB is parallel to the y-axis; for each of points A, B and C, both the x-coordinate and the y-coordinate must be integers; the range of possible x-coordinates is 0=<x=<5 and the range of possible y-coordinates is -4=<y=<6. Any two triangles with non-identical vertices are considered different. Given these constraints, how many different triangles could be constructed?

A. 50
B. 66
C. 2500
D. 3300
E. 4356
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Re: Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 14 May 2012, 23:47
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alexpavlos wrote:
Triangle ABC will be constructed in a xy-plane according to the following conditions: Angle ABC is 90 degrees, and AB is parallel to the y-axis; for each of points A, B and C, both the x-coordinate and the y-coordinate must be integers; the range of possible x-coordinates is 0=<x=<5 and the range of possible y-coordinates is -4=<y=<6. Any two triangles with non-identical vertices are considered different. Given these constraints, how many different triangles could be constructed?

A. 50
B. 66
C. 2500
D. 3300
E. 4356


Since give that 0=<x=<5 and -4=<y=<6 then we have a rectangle with dimensions 6*11 (6 horizontal and 11 vertical dots). AB is parallel to y-axis, BC is parallel to x-axis and the right angle is at B.

Choose the (x,y) coordinates for vertex B: 6C1*11C1;
Choose the x coordinate for vertex C (as y coordinate is fixed by B): 5C1, (6-1=5 as 1 horizontal dot is already occupied by B);
Choose the y coordinate for vertex A (as x coordinate is fixed by B): 10C1, (11-1=10 as 1 vertical dot is already occupied by B).

6C1*11C1*5C1*10C1=3,300.

Answer: D.

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Hope it helps.
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Re: Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 12 Feb 2017, 21:07
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Hi All,

While this prompt is a bit wordy (and even looks a bit 'complex), it can be broken down into small 'steps' rather easily.

We're given a number of 'restrictions' in terms of how a right triangle can drawn into an XY-plane:

1) The vertices must be comprised of integer values for the X and Y co-ordinates.
2) Segment AB is parallel to the Y-axis.
3) Angle B is the 90 degree angle.
4) The range of possible X co-ordinates is 0 <= X <= 5 and the range of possible Y-coordinates is -4 <= Y <= 6

We're asked to find the number of possible right triangles.

To start, let's place the first point ('Point A'). Considering the limits on the X and Y co-ordinates, there are 6 possible values for X and 11 possible values for Y. This means that there are (6)(11) = 66 possible places that we can place Point A.

Once we place Point A, we now have limitations on where we can place Point B. Since segment AB must be PARALLEL to the Y-axis, it has to have the SAME X-coordinate as Point A. Since we've placed Point A already, the Y-coordinate of Point B must be DIFFERENT from the one that Point A uses, so there are only 10 possibilities for Point B.

Once we place Point B, we now have limitations on where we can place Point C. Since angle B is a 90 degree angle, segment BC must be PARALLEL to the X-axis and Point C has to have the SAME Y-coordinate as Point B. Since we've placed Point B already, the X-coordinate of Point C must be DIFFERENT from the one that Point B uses, so there are only 5 possibilities for Point C.

In total, that gives us (66)(10)(5) = 3300 possible triangles.

Final Answer:

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Re: Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 27 Feb 2017, 21:43
This right triangle will have 3 points with the following coordinates : (x1,y1), (x1,y2), and (x2, y1)

Let (x1, y1) be the vertex with the right angle. x1 can have 6 different integer values. For each x1 there can be 11 different values that y1 can possibly take. Since this is a triangle x1 ~= x2 and y1~=y2. So for every x1, there is 5(6 minus 1) values that x2 can take. For every y1, there can be only 10 values that y2 can take. Hence, 6*5*11*10 =3300.

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Re: Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 20 Feb 2018, 21:15
Having a difficult time understanding the impact of this constraint: "Any two triangles with non-identical vertices are considered different." If this constraint was not given in the problem would the answer change?
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Re: Triangle ABC will be constructed in a xy-plane according to  [#permalink]

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New post 21 Feb 2018, 20:51
Carminaburana13 wrote:
Having a difficult time understanding the impact of this constraint: "Any two triangles with non-identical vertices are considered different." If this constraint was not given in the problem would the answer change?


Hi Carminaburana13,

In this question, we're drawing triangles by placing 3 co-ordinates on a graph. The 'intent' of this question is ultimately about finding 'unique' triangles, but if you're placing co-ordinates in different 'orders' then you could end up with multiple triangles in the exact SAME spot.

For example, consider the following triangle (based on the 'restrictions' in this prompt):
Point A (1,1), Point B (1,4) and Point C (5,4).

Depending on which co-ordinates you place first and second, you could end up with the following triangles that all share the SAME 3 co-ordinates:
ABC
ACB
BAC
BCA
CAB
CBA

This would make the correct answer 6 times larger (since each potential triangle would be counted 6 times instead of just once).

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Re: Triangle ABC will be constructed in a xy-plane according to &nbs [#permalink] 21 Feb 2018, 20:51
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