The coordinates of a quadrilateral are (5, 8), (7, 10), (9, 12), and (11, 14). after a transformation, the sign of the x-coordinate and the sign of the y-coordinate for each ordered pair have changed. which type of transformation occurred? a) 180° rotation b) (x, y)→(x + 2, y + 2) c) reflection across the y-axis d) reflection across the x-axis

A

Step-by-step explanation:

Under a rotation about the origin of 180°

a point (x, y ) → (- x, - y )

Under a reflection in the y- axis

a point (x, y ) → (- x, y )

Under a reflection in the x- axis

a point (x, y ) → (x, - y )

Since the sign of both coordinates have changed then the transformation is

A rotation of 180° → A

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.