Given P(1,-3); P'(-3,1) Q(3,-2);Q'(-2,3) R(3,-3);R'(-3,3) S(2,-4);S'(-4,2)
By observing the relationship between P and P', Q and Q',.... we note that (x,y)->(y,x) which corresponds to a single reflection about the line y=x. Alternatively, the same result may be obtained by first reflecting about the x-axis, then a positive (clockwise) rotation of 90 degrees, as follows: Sx(x,y)->(x,-y) [ reflection about x-axis ] R90(x,y)->(-y,x) [ positive rotation of 90 degrees ] combined or composite transformation R90. Sx (x,y)-> R90(x,-y) -> (y,x)
Similarly similar composite transformation may be obtained by a reflection about the y-axis, followed by a rotation of -90 (or 270) degrees, as follows: Sy(x,y)->(-x,y) R270(x,y)->(y,-x) => R270.Sy(x,y)->R270(-x,y)->(y,x)
So in summary, three ways have been presented to make the required transformation, two of which are composite transformations (sequence).
A random variable generator selects an integer from 1 to 100 both inclusive leaves us with total number of possible sample as 101.
We need to find the probability of selecting the integer 194.
The probability of selecting 194 from the sample is zero as the point does not exist in the random variable generator. Thus we can never pick 194 from the random variable generator giving us the probability a zero.
