Tough Coordinate Geo OG problem

[tonebeeze]

Right triangle PQR is to be constructed in the xy-plane
so that the right angle is at P and PR is parallel to the
x-axis. The x- and y-coordinates of P, Q, and R are to
be integers that satisfy the inequalities -4 ≤ x ≤ 5 and
6 ≤ y ≤ 16. How many different triangles with these
properties could be constructed?

(A) 110

(B) 1,100

(C) 9,900

(D) 10,000

(E) 12,100

What's the quickest method to solve this type of problem?

[Anurag@Gurome]

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x- and y-coordinates of P, Q, and R are to be integers that satisfy the inequalities -4 ≤ x ≤ 5 and
6 ≤ y ≤ 16. How many different triangles with these properties could be constructed?

(A) 110
(B) 1,100
(C) 9,900
(D) 10,000
(E) 12,100

x and y are integer and -4 ≤ x ≤ 5 and 6 ≤ y ≤ 16.
Therefore possible number of values of x is 10 and that of y is 11.

Now PR is parallel to x-axis => y coordinates of P and R must be same
As PQR is a right-angled triangle with right angle at P and PR parallel to x-axis, PQ must be parallel to y-axis. Which again infers x coordinates of P and Q must be same.

Thus we can freely choose the coordinates of P and according to that we have to select the coordinates of Q and R.

For P, (x, y) can be selected in --> x in 10 ways and y in 11 ways --> 10*11 ways
For Q, (x, y) can be selected in --> x is fixed and y in 10 ways (1 already selected for P) --> 10 ways
For R, (x, y) can be selected in --> x in 9 ways (1 already selected for P) and y is fixed --> 9 ways

Thus different coordinates of PQR can be simultaneously selected in 9*10*10*11 = 9900 ways. Each of these will result in a different right-angled triangle.

The correct answer is C.