Question

the vertices a(1, - 2), b(1, - 4), and c(5, - 2) form a triangle. the vertices a(-3, 1), b(-1, 1) and c(-3, 5) are the image of the triangle after a sequence of transformations. which sequence of transformations could produce the image from the pre - image?
a 90° clockwise rotation about the origin and then a reflection over the y - axis
a reflection over the x - axis and then a reflection over the y - axis
a reflection over the y - axis and then the 90° clockwise rotation about the origin
a 90° counterclockwise rotation about the origin and then a translation left 5 units

Explanation:

Step1: Apply transformation rules

• For \(A(1, - 2)\):
• Option 1: A reflection over the \(x -\)axis and then a reflection over the \(y -\)axis
• Reflection over \(x -\)axis: \((x,y)\to(x, - y)\), so \(A(1,-2)\to(1,2)\)
• Reflection over \(y -\)axis: \((x,y)\to(-x,y)\), so \((1,2)\to(-1,2)

eq A'(-3,1)\)

• Option 2: A \(90^{\circ}\) clockwise rotation about the origin and then a reflection over the \(y -\)axis
• \(90^{\circ}\) clockwise rotation about the origin: \((x,y)\to(y,-x)\), so \(A(1,-2)\to(-2,-1)\)
• Reflection over \(y -\)axis: \((x,y)\to(-x,y)\), so \((-2,-1)\to(2,-1)

eq A'(-3,1)\)

• Option 3: A \(90^{\circ}\) counter - clockwise rotation about the origin and then a translation left \(5\) units
• \(90^{\circ}\) counter - clockwise rotation about the origin: \((x,y)\to(-y,x)\), so \(A(1,-2)\to(2,1)\)
• Translation left \(5\) units: \((x,y)\to(x - 5,y)\), so \((2,1)\to(-3,1)\)
• For \(B(1,-4)\):
• \(90^{\circ}\) counter - clockwise rotation about the origin: \((x,y)\to(-y,x)\), so \(B(1,-4)\to(4,1)\)
• Translation left \(5\) units: \((x,y)\to(x - 5,y)\), so \((4,1)\to(-1,1)\)
• For \(C(5,-2)\):
• \(90^{\circ}\) counter - clockwise rotation about the origin: \((x,y)\to(-y,x)\), so \(C(5,-2)\to(2,5)\)
• Translation left \(5\) units: \((x,y)\to(x - 5,y)\), so \((2,5)\to(-3,5)\)

Answer:

A \(90^{\circ}\) counterclockwise rotation about the origin and then a translation left \(5\) units.