AQA GCSE Maths: Transformations and Vectors

$OACB$OACB is a parallelogram. $\overrightarrow{OA} = \mathbf{a}$OA=a and $\overrightarrow{OB} = \mathbf{b}$OB=b. The point $M$M is the midpoint of the diagonal $AB$AB.

(a) Write $\overrightarrow{AB}$AB in terms of $\mathbf{a}$a and $\mathbf{b}$b. (1 mark)

(b) Show that $\overrightarrow{OM} = \tfrac{1}{2}(\mathbf{a} + \mathbf{b})$OM=21​(a+b). (2 marks)

(c) The point $C$C is such that $OACB$OACB is a parallelogram, so $\overrightarrow{OC} = \mathbf{a} + \mathbf{b}$OC=a+b. Use your answer to part (b) to explain why the diagonals $OC$OC and $AB$AB of the parallelogram bisect each other. (2 marks)

Triangle $T$T has vertices at $(2, 1)$(2,1), $(4, 1)$(4,1) and $(4, 4)$(4,4). Triangle $T$T is enlarged by scale factor $-2$−2 with centre of enlargement $(1, 1)$(1,1) to give triangle $T'$T′.

(a) Work out the coordinates of the image of the vertex $(4, 4)$(4,4) 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 $\mathbf{p} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}$p=(5−2​) and $\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$q=(−14​).

(a) Work out $3\mathbf{p} - 2\mathbf{q}$3p−2q as a column vector. (2 marks)

(b) Work out the magnitude of $\mathbf{q}$q, giving your answer as a surd in its simplest form. (2 marks)

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

$\overrightarrow{AB} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$AB=(6−3​) and $\overrightarrow{BC} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$BC=(−15​). Work out $\overrightarrow{AC}$AC as a column vector. (2 marks)

Write down the column vector that describes the translation of the point $(2, 7)$(2,7) to the point $(9, 3)$(9,3). (1 mark)