Question

which explanation for the congruency of the trapezoids is correct? (1 point) the trapezoids are congruent because abcd was rotated 270° (counterclockwise). 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 90° (counterclockwise).

Explanation:

Step1: Recall rotation rules

For a 90 - degree counter - clockwise rotation of a point $(x,y)$ about the origin, the new point is $(-y,x)$. For a 270 - degree counter - clockwise rotation of a point $(x,y)$ about the origin, the new point is $(y, - x)$. For a reflection over the line $y = x$, the transformation of a point $(x,y)$ is $(y,x)$. For a translation left $a$ units and up $b$ units, a point $(x,y)$ becomes $(x - a,y + b)$.

Step2: Check rotation of 270 degrees

Take a point on trapezoid $ABCD$, say $A(2,1)$. After a 270 - degree counter - clockwise rotation about the origin, $A(2,1)$ would become $(1,-2)$ which does not match the position of $A'$.

Step3: Check reflection over $y = x$

If we reflect $A(2,1)$ over the line $y = x$, it becomes $(1,2)$ which does not match the position of $A'$.

Step4: Check translation

If we translate $A(2,1)$ left 4 units and up 2 units, $x=2,y = 1$, then the new point is $(2-4,1 + 2)=(-2,3)$ which is the position of $A'$. Checking other points: $B(7,1)$ translated left 4 units and up 2 units gives $(7 - 4,1+2)=(3,3)$ which is $B'$, $C(5,4)$ translated gives $(5 - 4,4 + 2)=(1,6)$ which is $C'$, $D(3,4)$ translated gives $(3 - 4,4 + 2)=(-1,6)$ which is $D'$.

Step5: Check 90 - degree rotation

For a 90 - degree counter - clockwise rotation of $A(2,1)$ about the origin, it becomes $(-1,2)$ which does not match the position of $A'$.

Answer:

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