Question

which sequences of transformations could map rectangle a onto rectangle b? select all the correct answers.
a a 90° clockwise rotation around the origin, followed by a translation to the left
b a dilation with a center of dilation at the origin, followed by a translation down and a translation to the left
c a reflection across the x - axis, followed by a dilation with a center of dilation at the origin and a translation to the left
d a reflection across the y - axis, followed by a translation down and a dilation with a center of dilation at the origin
e a 180° rotation around the origin, followed by a dilation with a center of dilation at the origin, a translation down, and a translation to the left

Explanation:

Step1: Recall transformation rules

Rotation, reflection, dilation and translation rules are key. A 90 - degree clock - wise rotation around the origin changes $(x,y)$ to $(y, - x)$. A dilation changes the size, a reflection flips the figure, and a translation moves it.

Step2: Analyze option A

A 90 - degree clockwise rotation around the origin will change the orientation of rectangle A. Then a left - translation can move it to a position similar to rectangle B's orientation and position. This is a possible sequence.

Step3: Analyze option B

A dilation with center at the origin changes the size of rectangle A. Then translations down and to the left can re - position it. This can map rectangle A onto rectangle B.

Step4: Analyze option C

A reflection across the x - axis flips rectangle A over the x - axis. A dilation changes its size and a left - translation moves it. This can map rectangle A onto rectangle B.

Step5: Analyze option D

A reflection across the y - axis flips rectangle A to the left - hand side of the y - axis. A down - translation and a dilation may not be able to map it onto rectangle B as the orientation will be wrong.

Step6: Analyze option E

A 180 - degree rotation around the origin flips rectangle A to the opposite quadrant. A dilation changes its size and translations down and to the left can re - position it. This can map rectangle A onto rectangle B.

Answer:

A. A 90° clockwise rotation around the origin, followed by a translation to the left
B. A dilation with a center of dilation at the origin, followed by a translation down and a translation to the left
C. A reflection across the x - axis, followed by a dilation with a center of dilation at the origin and a translation to the left
E. A 180° rotation around the origin, followed by a dilation with a center of dilation at the origin, a translation down, and a translation to the left