Question

rectangle a
rectangle b
which of the following sequences of transformations will carry rectangle a onto rectangle b?
a. rotate rectangle a 90° clockwise about the origin, and translate it 2 units up and 6 units to the right.
b. translate rectangle a 7 units down, rotate it 180° about the point (2, -6), and translate it 8 units to the left.
c. reflect rectangle a over the x - axis, translate it 2 units up, and reflect it over the y - axis.
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

For a 90 - degree clock - wise rotation about the origin, the transformation rule for a point \((x,y)\) is \((y, - x)\). Then translating 2 units up and 6 units right. Consider a vertex of rectangle A, say \((-2,3)\). After 90 - degree clock - wise rotation, it becomes \((3,2)\). After translation, it becomes \((3 + 6,2+2)=(9,4)\). But the corresponding vertex of rectangle B does not match.

Step2: Analyze Option B

Translate rectangle A 7 units down, a point \((x,y)\) becomes \((x,y - 7)\). Then rotate 180 - degree about the point \((2,-6)\). The formula for rotating a point \((x_1,y_1)\) 180 - degree about a point \((a,b)\) is \((2a - x_1,2b - y_1)\). Then translate 8 units left. Consider a vertex of rectangle A, say \((-2,3)\). After translating 7 units down, it is \((-2,-4)\). After rotating 180 - degree about \((2,-6)\), we have \(2\times2-(-2)=6\) and \(2\times(-6)-(-4)=-12 + 4=-8\). After translating 8 units left, we get \((6-8,-8)=(-2,-8)\). The corresponding vertex of rectangle B does not match.

Step3: Analyze Option C

Reflect rectangle A over the x - axis, the transformation rule for a point \((x,y)\) is \((x,-y)\). Then translate 2 units up \((x,-y + 2)\), and reflect over the y - axis \((-x,-y + 2)\). Consider a vertex of rectangle A, say \((-2,3)\). After reflecting over the x - axis, it is \((-2,-3)\). After translating 2 units up, it is \((-2,-1)\). After reflecting over the y - axis, it is \((2,-1)\). The corresponding vertex of rectangle B matches.

Step4: Analyze Option D

Rotate rectangle A 90 - degree counter - clockwise about the origin, the transformation rule for a point \((x,y)\) is \((-y,x)\). Then reflect over the y - axis \((y,x)\), and translate 1 unit up \((y,x + 1)\). Consider a vertex of rectangle A, say \((-2,3)\). After 90 - degree counter - clockwise rotation, it becomes \((-3,-2)\). After reflecting over the y - axis, it becomes \((3,-2)\). After translating 1 unit up, it becomes \((3,-1)\). The corresponding vertex of rectangle B does not match.

Answer:

C. Reflect rectangle A over the x - axis, translate it 2 units up, and reflect it over the y - axis.