OCR GCSE Maths: Transformations and Vectors

Triangle $P$P has vertices at $(2, 1)$(2,1), $(2, 3)$(2,3) and $(5, 1)$(5,1).

(a) Triangle $P$P is reflected in the line $y = -x$y=−x to give triangle $P'$P′. Work out the coordinates of the three vertices of triangle $P'$P′. (2 marks)

(b) Triangle $Q$Q has vertices at $(-2, -1)$(−2,−1), $(-2, -3)$(−2,−3) and $(-5, -1)$(−5,−1). Describe fully the single transformation that maps triangle $P$P onto triangle $Q$Q. (2 marks)

(c) Triangle $P$P is translated by the vector $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$(−32​). Write down the coordinates of the image of the vertex $(5, 1)$(5,1) under this translation. (1 mark)

Triangle $T$T has vertices at $(2, 2)$(2,2), $(2, 6)$(2,6) and $(4, 2)$(4,2). Triangle $T$T is enlarged by scale factor $-\tfrac{1}{2}$−21​ with centre of enlargement $(0, 0)$(0,0) to give triangle $T'$T′.

(a) Work out the coordinates of the image of the vertex $(2, 6)$(2,6) under this enlargement. (2 marks)

(b) State how the size and orientation of triangle $T'$T′ compare with triangle $T$T. (2 marks)

The vectors are $\mathbf{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$a=(4−3​) and $\mathbf{b} = \begin{pmatrix} -2 \\ 6 \end{pmatrix}$b=(−26​).

(a) Work out $2\mathbf{a} - \mathbf{b}$2a−b as a column vector. (2 marks)

(b) Work out the magnitude of $\mathbf{a}$a. (2 marks)

Triangle $R$R has vertices at $(1, 2)$(1,2), $(3, 2)$(3,2) and $(1, 5)$(1,5). Triangle $R$R is reflected in the line $x = 4$x=4 to give triangle $R'$R′. Write down the coordinates of the three vertices of triangle $R'$R′. (3 marks)

$\overrightarrow{AB} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$AB=(23​) and $\overrightarrow{CD} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$CD=(69​). Show that $AB$AB is parallel to $CD$CD. (2 marks)

Write down the column vector that describes the translation of the point $(4, 3)$(4,3) to the point $(-1, 8)$(−1,8). (1 mark)