Question

The image A"B"C" was created by dilating the pre-image by a scale factor of 1/3 about the origin,then reflected using the rule (x, y) - (-x, y). The coordinates of the image are A"(-2, -1) B"(0,3)and C"(4.1). Determine the coordinates of the preimage.

Answer 1

In order to find the coordinates of the preimage A, B and C, we need to go backwards in the transformations (that is, from A"B"C" to A'B'C', then from A'B'C' to ABC)

The tranformation from A'B'C' to A"B"C" is a reflection over the y-axis, that is, it causes a change in the x-coordinate signal.

So in order to find the coordinates of A'B'C', we need to undo this reflection:

`\begin{gathered} A^(\doubleprime)(-2,-1)\to A^(\prime)(2,-1) \\ B^(\doubleprime)(0,3)\to B^(\prime)(0,3) \\ C^(\doubleprime)(4,1)\to C^(\prime)(-4,1) \end{gathered}`

Now, the transformation from ABC to A'B'C' is a dilation by a scale factor of 1/3. In order to undo this, we need to apply the inverse dilation from A'B'C' to ABC, that is, a dilation by a scale factor of 3:

`\begin{gathered} A^(\prime)(2,-1)\to A(2\cdot3,-1\cdot3)=A(6,-3) \\ B^(\prime)(0,3)\to B(0\cdot3,3\cdot3)=B(0,9) \\ C^(\prime)(-4,1)\to C(-4\cdot3,1\cdot3)=C(-12,3) \end{gathered}`

So the coordinates of the preimage are A(6, -3), B(0, 9) and C(-12, 3).