Quote:
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
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