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.
_________________

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

Good question.+1 KUDO

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 coordinates of A, B and C are integers and satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. How many different triangles can be constructed with these properties?

A 63
B 336
C 567
D 3024
E 5040

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

All the cordinates are integer hence,
possible AB values are
{-1,0}, {-1,1} ....{-1,7}: 8 ways
....
{7,-1}, {7,0} ......{7,6}: 8 ways

9*8 ways = 72

for AC values can be
{1,2}, {1,3} .... {1,7} : 6ways
......
{7,1}, {7,2} ..... {7,6} : 6 ways

7*6 = 42

Total = 72 * 42 = 3024 hence D

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

Hope this helps
Cheers
J

Hi jlgdr... can u plz explain y the units digit has to 4... thanks

If we look at such questions as three slots that need to be filled with numbers, they become easier.

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

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

Analyze the Question from the perspective of each Point and the Possible Values it can take (also, I believe the question should mention only Integer Values are allowed....otherwise there would be too many too count)

We have 2 Major Constraints that we have to keep in mind:

(1)the Side AB must be Parallel to the X-Axis
and
(2)the 90 Degree Angle exists at Vertex A



1st) We'll Start with how many Coordinates we can Pick for A (X, Y)

for A's X Value, we have a Total of 9 Integer Options we can choose from [-1 thru 7] inclusive

for A's Y Value, we have a Total of 7 Integer Options we can choose from [+1 thru 7] inclusive


2nd) Moving to Point B

for B's Y Value: because the Side AB must be Parallel to the X-Axis, out of all the Different Arrangements of A Coordinates we can find, for any 1 of the Possibilities above the Y-Value of B MUST MATCH the Y-Coordinate of A. Only then will the Side AB be Parallel to the X-Axis.

Thus, there is only 1 Fixed Option for B's Y Coordinate that we can choose = whatever Y-Coord. is chosen for the A Point.


for B's X Value: So long as the X-Coord. of B does NOT = the X-Coord. of A, it can take on every other Integer Value allowed by the Constraints.

Thus there is --- [9 - 1] = 8 Options that can be chosen for B's X-Coord.


3rd)Finally, how may options do we have for Point C, given each distinct possibility of Point B and Point A above.

X-Coordinate of Point C: The Key Constraint is that the 90 Degree Angle must be at the A Vertex. Since Side AB must be Parallel to the X-Axis, this means that Vertex C must be Vertically Perpendicular to Point A.

once we have chosen an X-Coordinate for Point A out of all the Different Arrangements above, the X-Coordinate for Point C becomes FIXED. There is only 1 Option that C's X-Coord can take: must = A's X-Coord.

1 Option

Y-Coordinate of Point C: as long as Point C does NOT Share the Same Y-Coordinate as Point A, then it can take any Integer Value allowed within the Constraints

[7 - 1 Y-Coord. chosen for Point A] = 6 Options


Thus, using the Fundamental Counting Principle, The Total Number of Distinct Right Triangles that can be formed:


(9 * 7) * (8 * 1) * (1 * 6) = 63 * 48 = 3, 024 possibilities

-D-