George designs a computer program that can easily perform any number of transformations. To check his program, he enters the coordinates of the vertices of a quadrilateral as A(4, 5) theta(1, 2), c(3, 1) and D(6, 4) . He then types instructions to reflect the quadrilateral across the x-axis 99 times and then across the y-axis 99 times. Which figure shows the transformed quadrilateral generated by the program?

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.

you can't solve x because it is 2 linear equation and you only have one equation