Question

which transformations could have been used to create abc? choose two correct answers
a 180° rotation about the origin and then a translation 3 units up and 1 unit left
a 90° counterclockwise rotation about the origin and then a reflection over the x - axis
a 90° clockwise rotation about the origin and then a reflection over the y - axis
a translation 3 units down and 1 unit right and then a 180° rotation about the origin

Explanation:

Step1: Analyze the first transformation option

Let's assume the coordinates of points \(A\), \(B\), \(C\). Suppose \(A=(x_1,y_1)\), \(B=(x_2,y_2)\), \(C=(x_3,y_3)\).
For a \(180^{\circ}\) rotation about the origin, the transformation rule is \((x,y)\to(-x, -y)\). Then a translation \(3\) units up and \(1\) unit left: \((x,y)\to(x - 1,y + 3)\).

Step2: Analyze the second transformation option

For a \(90^{\circ}\) counter - clockwise rotation about the origin, the transformation rule is \((x,y)\to(-y,x)\). Then a reflection over the \(x\) - axis: \((x,y)\to(x,-y)\).

Step3: Analyze the third transformation option

For a \(90^{\circ}\) clockwise rotation about the origin, the transformation rule is \((x,y)\to(y,-x)\). Then a reflection over the \(y\) - axis: \((x,y)\to(-x,y)\).

Step4: Analyze the fourth transformation option

For a translation \(3\) units down and \(1\) unit right: \((x,y)\to(x + 1,y-3)\). Then a \(180^{\circ}\) rotation about the origin: \((x,y)\to(-x,-y)\).

By using the transformation rules for rotation (\(90^{\circ}\) clockwise \((x,y)\to(y, - x)\) followed by reflection over \(y\) - axis \((x,y)\to(-x,y)\)) and \(180^{\circ}\) rotation about the origin followed by translation, we can verify the correct transformations.

Let’s assume \(A=(1,-4)\), \(B=(5,-5)\), \(C=(4,-2)\) (original triangle \(A'B'C'\)) and for the upper - left triangle assume \(A=( - 1,-4)\), \(B=( - 5,-5)\), \(C=( - 4,-2)\) (before translation).

• For \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) and then translation \(3\) units up and \(1\) unit left \((x,y)\to(x - 1,y + 3)\):

If we start with a point \((x,y)\) in \(A'B'C'\), after \(180^{\circ}\) rotation \((-x,-y)\), after translation \((-x - 1,-y + 3)\)

• For \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y,-x)\) and then reflection over \(y\) - axis \((x,y)\to(-x,y)\)

Answer:

a \(180^{\circ}\) rotation about the origin and then a translation \(3\) units up and \(1\) unit left; a \(90^{\circ}\) clockwise rotation about the origin and then a reflection over the \(y\) - axis