since edge is on x axis, two vertices will be on x axis that is \(( x_1,0)\) and \((x_2,0)\) and x can take any of the 4 values - 0,1,2,3

I. total triangles possible = choose any two of 0,1,2,3 so 4C2=6
and for this the vertex can be any of the 3*4=12, 3 for y= 1,2, and 3 and 4 for x as 0,1,2,3
total 12*6=72


II. when area = integer
for this one of the side is EVEN as area = \(\frac{1}{2} * x*y\)
when the edge on x axis is odd ...
(0,0) and (1,0); (1,0) and (2,0); (2,0) and (3,0); (0,0) and (3,0); so 4 cases
y has to be even so 2 so (0,2), (1,2),(2,2),(3,2) so 4 cases
total 4*4=16
when the edge on x axis is even ...
(0,0) and (2,0); (1,0) and (3,0); so 2 cases
y can be anything so 2 so 1,2 and 3 and x 0,1,2,3 so 3*4 = 12 cases
total 2*12=24
Overall = 16+24=40

prob = \(\frac{40}{72}=\frac{5}{9}\)

hi chetan2u, nice explanation though i do have questions if you dont mind

can you please help me to understand your solution ... i definately dont understand one thing how can we create a triangle with the coordinates in which

x or y is 0 for instance you wrote these coordinates, see in red (0,0) and (1,0); (1,0) and (2,0); (2,0) and (3,0); (0,0) and (3,0); so 4 cases
y has to be even so 2 so (0,2), (1,2),(2,2),(3,2) so 4 cases


see attached diagram this is how i understood the problem as you see for one of the edges of triangles to be on X axis i got folowing options (3:3), (2:3) (2;2) , (2:1), (1:1)

Also i didnt get this "and for this the vertex can be any of the 3*4=12, 3 for y= 1,2, and 3 and 4 for x as 0,1,2,3
total 12*6=72" can you please somehow rephrase the explanation the only thing i got is that triangle has 3 vertices ....

And one more question how did you manage to solve it without visualizing the problem ?

i would appreciate your taking time to explain thank you!

how is it possible to draw a triangle with these coordinates (0,0) and (1,0) ? can someone draw such triangle ?

Hi..
Now coordinate(0,0) & (1,0) will give two vertices say A and B now C can be any of the 12 points above. So C can be (0,1) or (1,1) or (2,1), or (3,1) or (0,2) or (1,2) and so on and all these C coordinates will give different triangles
Similarly for (0,0) and (0,2) , again C can be any of the coordinates as shown above

thank you got it ! could you please rephrase the following sentence if you don't mind "and for this the vertex can be any of the 3*4=12, 3 for y= 1,2, and 3 and 4 for x as 0,1,2,3 total 12*6=72"


just need to understand it

Hi chetan2u

If you don't mind could you pls explain a bit more how you are calculating the total number of triangles.

Thank you in advance.

Hi Chetan,
Can you please elaborate on the line " for this the vertex can be any of the 3*4=12, 3 for y= 1,2, and 3 and 4 for x as 0,1,2,3
total 12*6=72".

I am nt very clear how you got 12 combinations for those 6 ways
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