geometry n combinatorics

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 ?

P(x, y)
R(x+k, y)
Q(x, y+m)

Select x and y, then the number of choices for x+k and y+k would be #x - 1 and #y -1

–4 ≤ x ≤ 5 = 5-(-4)+1 = 10
6 ≤ y ≤ 16 = 16-6+1 = 11

Ans: 10*11*9*10 = 9900

Variations of the same question:



Number of rectangles with that restriction (PR || x-axis) = Number of right angled triangles, since once ya fix PQR, the another vertex is unique = 9900

Number of quadrilaterals should be less than (11*10) C 4

Quadrilateral inequality