Question

which sequence of transformations produces abc from abc? a translation up 3 and then a 90° counterclockwise rotation about the origin a 90° counterclockwise rotation about the origin and then a translation up 3 a translation left 5 and down 2 and then a reflection over the x - axis a reflection over the x - axis and then a translation left 2 and down 1

Explanation:

Step1: Analyze the coordinates

Let's assume the coordinates of the vertices of \(\triangle ABC\) are \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\) and of \(\triangle A'B'C'\) are \(A'(x_1',y_1')\), \(B'(x_2',y_2')\), \(C'(x_3',y_3')\).
If we consider a reflection over the \(x -\)axis, the transformation rule is \((x,y)\to(x, - y)\). Then for a translation \((x,y)\to(x - 2,y-1)\) (left \(2\) units and down \(1\) unit).
Suppose \(A\) has coordinates \((a,b)\). After reflection over the \(x -\)axis, it becomes \((a,-b)\). After translation left \(2\) and down \(1\), it becomes \((a - 2,-b - 1)\).

Step2: Check other options

• For a \(90^{\circ}\) counter - clockwise rotation about the origin, the transformation rule is \((x,y)\to(-y,x)\). Then a translation up \(3\) would be \((x,y)\to(-y,x + 3)\), which does not match the general pattern of the given transformation.
• For a translation left \(5\) and down \(2\) \((x,y)\to(x - 5,y-2)\) and then a reflection over the \(x -\)axis \((x,y)\to(x - 5,-(y - 2))=(x - 5,-y + 2)\), which does not match the given transformation.
• For a translation up \(3\) and then a \(90^{\circ}\) counter - clockwise rotation about the origin, using the rotation rule \((x,y)\to(-y,x)\) after translation \((x,y)\to(x,y + 3)\) gives \((x,y)\to(-(y + 3),x)\), which does not match the given transformation.

Answer:

a reflection over the \(x -\)axis and then a translation left \(2\) and down \(1\)