WAEC 2018 · Paper 2 · Q13

Vectors & transformations

(a)
Using a scale of 2 cm to 2 units on both axes, draw on a sheet of graph paper two perpendicular axes $Ox$ and $Oy$ for $-10 \le x \le 10$ and $-10 \le y \le 10$ .
Model answer
Draw both axes with 2 cm to 2 units from $-10$ to $10$ , and label them. Then:
- $PQRS$ : $P(3, 2)$ , $Q(-1, 5)$ , $R(0, 8)$ , $S(3, 7)$ .
- Rotation of $90^\circ$ anticlockwise about $O$ : $(x, y) \to (-y, x)$ , giving $P_1(-2, 3)$ , $Q_1(-5, -1)$ , $R_1(-8, 0)$ , $S_1(-7, 3)$ .
- Reflection in $y = x$ : $(x, y) \to (y, x)$ , giving $P_2(3, -2)$ , $Q_2(-1, -5)$ , $R_2(0, -8)$ , $S_2(3, -7)$ .
$P_2Q_2R_2S_2$ is $PQRS$ flipped in the $x$ -axis, which is the answer to (c).
(b)
Given the points $P(3, 2)$ , $Q(-1, 5)$ , $R(0, 8)$ and $S(3, 7)$ , draw on the same graph, indicating clearly the vertices and their coordinates, the: (i) quadrilateral $PQRS$ ; (ii) image $P_1Q_1R_1S_1$ of $PQRS$ under an anticlockwise rotation of $90^\circ$ about the origin; (iii) image $P_2Q_2R_2S_2$ of $P_1Q_1R_1S_1$ under a reflection in the line $y - x = 0$ .
Model answer
The rotation maps $(x, y)$ to $(-y, x)$ : $P_1(-2, 3)$ , $Q_1(-5, -1)$ , $R_1(-8, 0)$ , $S_1(-7, 3)$ . Reflecting in $y = x$ swaps the coordinates: $P_2(3, -2)$ , $Q_2(-1, -5)$ , $R_2(0, -8)$ , $S_2(3, -7)$ , which is $PQRS$ reflected in the $x$ -axis.
(c)
Describe precisely the single transformation $T$ for which $T : PQRS \to P_2Q_2R_2S_2$ .
Show the answer
A reflection in the $x$ -axis
(d)
The side $P_1Q_1$ of the quadrilateral $P_1Q_1R_1S_1$ cuts the $x$ -axis at the point $W$ . What type of quadrilateral is $P_1S_1R_1W$ ?
Show the answer
A trapezium ( $P_1S_1 \parallel R_1W$ )

Try it on a graph

PQRS (blue), P₁Q₁R₁S₁ (red), P₂Q₂R₂S₂ (green).

Worked solution (try it first)

(a)

With 2 cm to 2 units, draw both axes from $-10$ to $10$ , crossing at $O$ , and number them.

(b)(i)

Plot $P(3, 2)$ , $Q(-1, 5)$ , $R(0, 8)$ and $S(3, 7)$ and join them in order.

(ii)

A rotation of $90^\circ$ anticlockwise about the origin sends $(x, y)$ to $(-y, x)$ : $P_1(-2, 3)$ , $Q_1(-5, -1)$ , $R_1(-8, 0)$ and $S_1(-7, 3)$ .

(iii)

$y - x = 0$ is the line $y = x$ .
A reflection in it swaps the coordinates, $(x, y) \to (y, x)$ : $P_2(3, -2)$ , $Q_2(-1, -5)$ , $R_2(0, -8)$ and $S_2(3, -7)$ .

(c)

Compare each point with its final image: $P(3, 2) \to P_2(3, -2)$ and $Q(-1, 5) \to Q_2(-1, -5)$ .
Each $(x, y)$ goes to $(x, -y)$ , so $T$ is a reflection in the $x$ -axis.

(d)

$P_1(-2, 3)$ and $Q_1(-5, -1)$ : the gradient of $P_1Q_1$ is $\frac{-1 - 3}{-5 - (-2)} = \frac43$ .
The line is $y - 3 = \frac43(x + 2)$ , and $y = 0$ gives $x + 2 = -\frac94$ , so $W$ is $(-4.25, 0)$ .
$P_1S_1$ lies on $y = 3$ and $R_1W$ on $y = 0$ , so they are parallel.
The other two sides are not parallel (gradients 3 and $\frac43$ ), so $P_1S_1R_1W$ is a trapezium.

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