Raksat wrote:
From among all the triangles that could be drawn in the coordinate plane with vertices with integer coordinates (x,y) satisfying 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3, one triangle is to be chosen at random. If one of the edges of this triangle lies on the x-axis, what is the probability the area of the triangle is an integer ?
a) 1/3
B)1/2
C)5/9
D)2/3
E)3/4
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=16when 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=24Overall = 16+24=40
prob = \(\frac{40}{72}=\frac{5}{9}\)