
Right triangle PQR is to be constructed in the xy– plane so that the right angle is at P and line PR is parallel to theaxis. 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 could be constructed with these properties?
 110
 1,100
 9,900
 10,000
Correct Option : C
We have the rectangle with dimensions 10 ×11 (10 horizontal dots and 11 vertical). PQ is parallel to yaxis and PR is parallel to xaxis.
Choose the (x, y) coordinates for vertex P (right angle): ^{10}C_{1} × ^{11}C_{1}
Choose the x coordinate for vertex R (as y coordinate is fixed by A): ^{8}C1, (10 – 1 = 9 as 1 horizontal dot is already occupied by A)
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): ^{10}C_{1}, (11 – 1 = 10 as 1 vertical dot is already occupied by A).
Hence, required number of triangles will be ^{10}C_{1} × ^{11}C_{1} × ^{9}C_{1} × ^{10}C_{1} = 9900.
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