Negative-scale-factor enlargement

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 4 marks
The point $P(4, 3)$ is enlarged by scale factor $-2$ with centre $C(1, 1)$ .
(a) Find the coordinates of the image of $P$ . (2)
(b) Describe fully the single transformation that maps the image back onto $P$ . (2)
Question 2Non-calculator · 5 marks
An enlargement maps A $(6,3)$ to A′ $(-4,-7)$ with scale factor $-3/2$ .
(a) Find the centre of enlargement. (3)
(b) A side has original length 8 cm. Find its image length. (2)
Question 3Non-calculator · 3 marks
Transformation $E$ is an enlargement, scale factor $-2$ , centre $(0, 0)$ . Transformation $F$ is an enlargement, scale factor $-\frac{1}{2}$ , centre $(3, 0)$ . A shape is transformed by $E$ and then by $F$ .
(a) Show that the combined transformation is a translation, and find its vector. (3)

Answers and marks

(a) $(-5, -3)$

(b) Enlargement, scale factor $-\frac{1}{2}$ , centre $(1, 1)$

(a) $(2,-1)$

P1 Let the centre be (c, d). For each coordinate, image = centre + scale factor × (original − centre).
P1 Apply the same relationship to the vertical coordinate.
A1 Correct answer: $(2,-1)$

(b) $12$ cm

(a) Translation by $\begin{pmatrix} 4.5 \\ 0 \end{pmatrix}$

M1 Applying $E$ to a general point: $(-2x, -2y)$ .
M1 Applying $F$ to their image, with the centre $(3, 0)$ used correctly.
A1 Showing the image is $(x + 4.5, y)$ and concluding it is a translation by the vector with components 4.5 and 0.

Negative-scale-factor enlargement

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