Question

which of the following sequences of transformations will carry rectangle a onto rectangle b?
a. reflect rectangle a over the x - axis, translate it 2 units up, and reflect it over the y - axis
b. rotate rectangle a 90° clockwise about the origin, and translate it 2 units up and 6 units to the right
c. translate rectangle a 7 units down, rotate it 180° about the point (2, 6), and translate it 6 units to the left
d. rotate rectangle a 90° counterclockwise about the origin, reflect it over the y - axis, and translate it 1 unit up

Explanation:

Step1: Analyze Option A

• Reflecting over the \(x -\)axis changes the \(y -\)coordinate sign. Translating 2 units up and reflecting over the \(y -\)axis won't map rectangle A to B.

Step2: Analyze Option B

• Rotating \(90^{\circ}\) clockwise about the origin changes the orientation. Translating 2 units up and 6 units right won't map rectangle A to B.

Step3: Analyze Option C

• Translating 7 units down, rotating \(180^{\circ}\) about \((2,6)\) (which is a valid rotation - translation combination for such a figure mapping), and translating 6 units to the left can map rectangle A to B.

Step4: Analyze Option D

• Rotating \(90^{\circ}\) counter - clockwise about the origin changes the orientation. Reflecting over the \(y -\)axis and translating 1 unit up won't map rectangle A to B.

Answer:

C. Translate rectangle A 7 units down, rotate it \(180^{\circ}\) about the point \((2,6)\), and translate it 6 units to the left.