A right triangle AQC has to be constructed in the xy-plane so that the right angle is at A and AQ is parallel to x axis. The coordinates of A, Q 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?  Select the correct option:
A) 63
B) 336
C) 567
D) 3024
E) 5040
All the cordinates are integer hence,   possible AQ values are   {-1,0}, {-1,1} ....{-1,7}: 8 ways   ....   {7,-1}, {7,0} ......{7,6}: 8 ways     9*8= 
<gadget id="calculator">  9*8</gadget>
<output>72</output>
72   for AC values can be   {1,2}, {1,3} .... {1,7} : 6ways   ......   {7,1}, {7,2} ..... {7,6} : 6 ways     7*6= 
<gadget id="calculator">  7*6</gadget>
<output>42</output>
42   Total= 72 * 42= 
<gadget id="calculator">72 * 42</gadget>
<output>3_024</output>
3024 hence 3024
<result>D</result>
A right triangle AQC has to be constructed in the xy-plane so that the right angle is at A and AQ is parallel to x axis. The coordinates of A, Q 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?