WAEC 2016 · Paper 2 · Q13

1. (a)

   Using a scale of 2 cm to 2 units on each axis, draw on a sheet of graph paper two perpendicular axes $Ox$ and $Oy$ for $-10 \le x \le 10$ and $-12 \le y \le 12$.

   Model answer

   Draw both axes with 2 cm to 2 units, from $-10$ to $10$ across and $-12$ to $12$ up, and label them. Then plot and join each shape:

   • $PQRS$: $P(-5, -4)$, $Q(2, -1)$, $R(0, 3)$, $S(-8, 4)$.
   • Translation by $\begin{pmatrix} 3 \\ -8 \end{pmatrix}$ (add 3 to $x$, subtract 8 from $y$): $P_1(-2, -12)$, $Q_1(5, -9)$, $R_1(3, -5)$, $S_1(-5, -4)$.
   • Enlargement, scale factor $-\frac12$ from $O$ (halve each coordinate and change its sign): $P_2(2.5, 2)$, $Q_2(-1, 0.5)$, $R_2(0, -1.5)$, $S_2(4, -2)$. The image is upside down and on the other side of $O$.

   Notice $S_1$ lands exactly on $P$.

2. (b)

   Draw on this graph, indicating the coordinates of all vertices: (i) the quadrilateral $PQRS$ with vertices $P(-5, -4)$, $Q(2, -1)$, $R(0, 3)$ and $S(-8, 4)$; (ii) the image $P_1Q_1R_1S_1$ of $PQRS$ under a translation by the vector $\begin{pmatrix} 3 \\ -8 \end{pmatrix}$; (iii) the image $P_2Q_2R_2S_2$ of $PQRS$ under an enlargement from the origin with scale factor $-\frac12$.

   Model answer

   Plot and join each shape, labelling every vertex with its coordinates: $PQRS$ with $P(-5, -4)$, $Q(2, -1)$, $R(0, 3)$, $S(-8, 4)$; the translation $P_1(-2, -12)$, $Q_1(5, -9)$, $R_1(3, -5)$, $S_1(-5, -4)$ (dashed; $S_1$ lands on $P$); and the enlargement with scale factor $-\frac12$ from $O$, $P_2(2.5, 2)$, $Q_2(-1, 0.5)$, $R_2(0, -1.5)$, $S_2(4, -2)$ (highlighted), which is upside down on the other side of $O$.

3. (c)

   Find the equation of the line $P_1S$.

   Show the answer

   $8x + 3y + 52 = 0$

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