Find image coordinates under transformation

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).

2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).

1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4 \\ - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
3. find the value of \(a\).

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point \(( 10,50 )\) under transformation T .
2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
6. Find the \(2 \times 2\) matrix which represents the transformation.
7. Show that this matrix is singular. Relate this result to the transformation.

1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).

The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$

1. Find the coordinates of the vertices of \(T ^ { \prime }\)
2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
4. Find the matrix \(\mathbf { R Q }\)
5. Give a full geometrical description of the single transformation represented by the answer to part (d).

3 Fig. 3 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { P }\) representing this transformation.
2. The parallelogram \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is transformed by the matrix \(\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\). Find the coordinates of the vertices of its image, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\), following this transformation.
3. Describe fully the transformation represented by \(\mathbf { Q P }\).

9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.

1. Describe fully the transformations P and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
   Triangle T is transformed first by transformation P , then by transformation Q .
2. Find the single matrix that represents this composite transformation.
3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.

9

1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1 \\ - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1 \\ 0 & 1 \end{array} \right)\).
2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).

9 The triangle ABC has vertices at \(\mathrm { A } ( 0,0 ) , \mathrm { B } ( 0,2 )\) and \(\mathrm { C } ( 4,1 )\). The matrix \(\left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right)\) represents a transformation T .

1. The transformation \(T\) maps triangle \(A B C\) onto triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Find the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\). Triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\) is now mapped onto triangle \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\) using the matrix \(\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right)\).
2. Describe fully the transformation represented by \(\mathbf { M }\).
3. Triangle \(\mathrm { A } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\) is now mapped back onto ABC by a single transformation. Find the matrix representing this transformation.
4. Calculate the area of \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\).

14 The matrix \(\mathbf { M }\) represents the transformation T , and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 & - 1 \\ - 2 & 6 \end{array} \right]$$ 14

1. The point \(A\) has coordinates ( \(4 , - 5\) )
   Find the coordinates of the image of \(A\) under T
   14
2. Show that the only invariant point under T is the origin.
   14
3. The line \(L _ { 1 }\) has equation \(y = x + 1\) The transformation \(T\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\) Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\)

A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.

1. Write down the image of the point \((2, 3)\) under this transformation. [2]
2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]

The matrix **M** is given by \(\mathbf{M} = \begin{pmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{pmatrix}\).

1. The diagram in the Printed Answer Booklet shows the unit square \(OABC\). The image of the unit square under the transformation represented by **M** is \(OA'B'C'\). Draw and clearly label \(OA'B'C'\). [3]
2. Find the equation of the line of invariant points of this transformation. [3]
3. 1. Find the determinant of **M**. [1]
   2. Describe briefly how this value relates to the transformation represented by **M**. [2]