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 right triangle's longest side is the hypotenuse let x=longest, y=middle, and z=shortest x=y+2 y=2z-1 therefore x=(2z-1)+2=2z+1 find z z^2+y^2=x^2 by Pythagorean theorem plug in x and y in terms of z z^2+(2z-1)^2=(2z+1)^2 z^2+4z^2-4z+1=4z^2+4z+1 subtract the right-hand side's value from the left-hand side's z^2-8z=0 z(z-8)=0 z=0, 8 z cannot be zero as the sides must have some value to it. Therefore the shortest side is equal to 8
