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

Question

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

A. 63
B. 336
C. 567
D. 3024
E. 5040

Bunuel · Accepted Answer

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.

cabelk · Answer

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

sudhir18n · Answer

Its D ,
for point A you can use any of 9 points ( Xaxis) and & 7 points ( y axis points).. hence 63 ways
for B he can use any other 8 points on the X axis 
and for C .. any of the other 6 points.
so total 63*8*6 = 3024

jlgdr · Answer

Units digit has to be 4 to quicken answer choice. 9*7*8*6 = 3024

Hope this helps
Cheers
J

ravihanda · Answer

Here we have a rectangle of size 9*7.
There are 9 choices for the x coordinate {-1,0,1,2,3,4,5,6,7}
There are 7 choices for the y coordinate {1,2,3,4,5,6,7}
AB is parallel to the x-axis, so both A & B will have the same y-coordinate. Let us say that is p
AC is parallel to the y-axis, so both A & C will have the same x-coordinate. Let us say that is q
Let us assume the y coordinate of x coordinate of B to be r and the y-coordinate of C be s
So, the coordinates of A are (q,p); B are (r,p); and C are (q,s)

We need to find out overall possible values for p,q,r,s

q and r need to be chosen from the 9 choices for x-coordinate
=> q & r can be chosen in 9*8 = 72 ways

p and s need to be chosen from the 8 choices for y-coordinate
=> p & s can be chosen in 7*6 = 42 ways
 
Total ways = 72*42 = 3024 
Correct Answer: D

tomcruiseh · Answer

Its not given that x,y has to be integer values only.

udaypratapsingh99 · Answer

1) Select the x coordinate of 2 vertices parallel to x-axis from -1 to 7: 9P2 = \frac{9!}{(9-2)!}= 9*8 ways
2) Select the y coordinate of A: 7 ways
3) Select the y coordinate of the 3rd point: 6 ways

Required answer = 9*8*7*6 = 3024

D is the correct answer

udaypratapsingh99 · Answer

You're right  tomcruiseh

bb 
 Bunuel  
can you please check this? I think it's a typo.

Bunuel · Answer

Fixed that. Thank you for reporting.

bumpbot · Answer

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

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