Question

12 the diagram shows shape x. make a copy of
the diagram.
a draw the image of x after a reflection in the line
x = -2, followed by a rotation of 90° clockwise
about the point (-1, 0), followed by a translation
(4
3), followed by a reflection in the line y = 1,
followed by a rotation of 90° anticlockwise about
the point (5, -1). label the image y.
b describe the single transformation that takes y to x.

Explanation:

Step1: Reflection in \(x = - 2\)

For a point \((x,y)\) reflected in the line \(x = a\), the formula is \((2a - x,y)\). Here \(a=-2\), for a general point \((x,y)\) its image is \((-4 - x,y)\).

Step2: Rotation of \(90^{\circ}\) clockwise

The rotation matrix for a \(90^{\circ}\) clockwise rotation about the origin is \(

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

\). If we have a point \((x,y)\), after rotation it becomes \((y,-x)\).

Step3: Translation

A translation \(

$$\begin{pmatrix}h\\k\end{pmatrix}$$

\) moves a point \((x,y)\) to \((x + h,y + k)\). Here \(

$$\begin{pmatrix}3\\4\end{pmatrix}$$

\), so \((x,y)\to(x + 3,y + 4)\).

Step4: Reflection in \(y = 1\)

For a point \((x,y)\) reflected in the line \(y=b\), the formula is \((x,2b - y)\). Here \(b = 1\), so \((x,y)\to(x,2 - y)\).

Step5: Rotation of \(90^{\circ}\) anticlockwise

The rotation matrix for a \(90^{\circ}\) anticlockwise rotation about the origin is \(

$$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$

\). If we have a point \((x,y)\), after rotation it becomes \((-y,x)\).

Answer:

To draw the image in part (a), follow the sequence of transformations: reflection in \(x=-2\), then \(90^{\circ}\) clockwise rotation, then translation \(

$$\begin{pmatrix}3\\4\end{pmatrix}$$

\), then reflection in \(y = 1\). For part (b), since the composition of transformations is a sequence of isometries (reflections, rotations, translations), we can work backwards. The single transformation that takes \(Y\) to \(X\) is a rotation of \(90^{\circ}\) clockwise (the inverse of the \(90^{\circ}\) anticlockwise rotation in the reverse - order of transformations).