Question

which explanation for the congruency of the trapezoids is correct? note be sure to acknowledge which figure is the original (1 point) the trapezoids are congruent because abcd was reflected over the line y = x the trapezoids are congruent because abcd was translated left 4 units and up 2 units the trapezoids are congruent because abcd was rotated 270 degrees counterclockwise the trapezoids are congruent because abcd was rotated 90 degrees counterclockwise

Explanation:

Step1: Analyze translation

For a translation, we check the change in x - and y - coordinates of corresponding points. If we consider point A(3,1) in trapezoid ABCD and point A'(- 1,3) in trapezoid A'B'C'D', the x - coordinate of A changes from 3 to - 1 (a change of 3-( - 1)=4 units to the left) and the y - coordinate of A changes from 1 to 3 (a change of 3 - 1 = 2 units up).

Step2: Check other points

For point D(3,4) in trapezoid ABCD and point D'(-1,6) in trapezoid A'B'C'D', the x - coordinate changes from 3 to - 1 (4 units left) and the y - coordinate changes from 4 to 6 (2 units up). This is consistent with a translation of left 4 units and up 2 units for all corresponding points.

Step3: Analyze reflection over y = x

The rule for reflection over the line y = x is (x,y)→(y,x). For example, if we take point A(3,1), after reflection over y = x, it should be (1,3) which is not the case for A'(-1,3).

Step4: Analyze rotation 270° counter - clockwise

The rule for a 270° counter - clockwise rotation about the origin is (x,y)→(y, - x). For point A(3,1), after 270° counter - clockwise rotation, it should be (1, - 3) which is not A'(-1,3).

Step5: Analyze rotation 90° counter - clockwise

The rule for a 90° counter - clockwise rotation about the origin is (x,y)→(-y,x). For point A(3,1), after 90° counter - clockwise rotation, it should be (-1,3), but checking other points shows it is not consistent for the whole trapezoid.

Answer:

The trapezoids are congruent because ABCD was translated left 4 units and up 2 units.