Question

the vertices a(-2, -1), b(-3, 2), c(-1, 3), and d(0, 0) form a parallelogram. the vertices a(-1, -2), b(2, -3), c(3, -1), and d(0, 0) are the image of the parallelogram after a sequence of transformations. which sequence of transformations could produce the image from the pre - image?

1. a reflection over the y - axis and then a 90° clockwise rotation about the origin
2. a reflection over the x - axis and then a reflection over the y - axis
3. a 90° clockwise rotation about the origin and then a reflection over the y - axis
4. a 90° counterclockwise rotation about the origin and then a reflection over the x - axis

Explanation:

Step1: Recall rotation and reflection rules

• Rotation rules:
• For a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\).
• For a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\).
• Reflection rules:
• Reflection over the \(x\) - axis: \((x,y)\to(x, - y)\).
• Reflection over the \(y\) - axis: \((x,y)\to(-x,y)\).

Step2: Analyze the first option (a reflection over the \(y\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin)

• Let's take point \(A(-2,-1)\).
• First, reflection over the \(y\) - axis: \((-2,-1)\to(2,-1)\).
• Then, \(90^{\circ}\) clockwise rotation about the origin: \((2,-1)\to(-1,-2)\) (which is \(A'\)).
• Take point \(B(-3,2)\).
• First, reflection over the \(y\) - axis: \((-3,2)\to(3,2)\).
• Then, \(90^{\circ}\) clockwise rotation about the origin: \((3,2)\to(2,-3)\) (which is \(B'\)).
• Take point \(C(-1,3)\).
• First, reflection over the \(y\) - axis: \((-1,3)\to(1,3)\).
• Then, \(90^{\circ}\) clockwise rotation about the origin: \((1,3)\to(3,-1)\) (which is \(C'\)).
• Take point \(D(0,0)\).
• First, reflection over the \(y\) - axis: \((0,0)\to(0,0)\).
• Then, \(90^{\circ}\) clockwise rotation about the origin: \((0,0)\to(0,0)\) (which is \(D'\)).

Step3: Analyze the second option (a reflection over the \(x\) - axis and then a reflection over the \(y\) - axis)

• Let's take point \(A(-2,-1)\).
• First, reflection over the \(x\) - axis: \((-2,-1)\to(-2,1)\).
• Then, reflection over the \(y\) - axis: \((-2,1)\to(2,1)

eq A'(-1,-2)\).

Step4: Analyze the third option (a \(90^{\circ}\) clockwise rotation about the origin and then a reflection over the \(y\) - axis)

• Let's take point \(A(-2,-1)\).
• First, \(90^{\circ}\) clockwise rotation about the origin: \((-2,-1)\to(-1,2)\).
• Then, reflection over the \(y\) - axis: \((-1,2)\to(1,2)

eq A'(-1,-2)\).

Step5: Analyze the fourth option (a \(90^{\circ}\) counter - clockwise rotation about the origin and then a reflection over the \(x\) - axis)

• Let's take point \(A(-2,-1)\).
• First, \(90^{\circ}\) counter - clockwise rotation about the origin: \((-2,-1)\to(1,-2)\).
• Then, reflection over the \(x\) - axis: \((1,-2)\to(1,2)

eq A'(-1,-2)\).

Answer:

The first option (a reflection over the \(y\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin)