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Translations and dilations

Translate shapes using vectors and dilate shapes from the origin using a scale factor.

Worked examples

Translating a point using a vector

Straightforward

Problem

Translate point A(2,5)A(2, 5) by the vector (3−4)\begin{pmatrix} 3 \\ -4 \end{pmatrix} and state the coordinates of A′A'.

Dilating a point from the origin

Moderate

Problem

Point B(4,−2)B(4, -2) is dilated from the origin by scale factor 33. State the coordinates of B′B'.

Finding the scale factor and classifying a dilation

Challenging

Problem

Point C(5,10)C(5, 10) is mapped to C′(2,4)C'(2, 4) by a dilation from the origin. Find the scale factor and classify the transformation as an enlargement, reduction or congruent.

Practise

Q1·Straightforward
Point A(3,2)A(3, 2) is translated by the vector (4−1)\begin{pmatrix} 4 \\ -1 \end{pmatrix}. State the coordinates of A′A'.
Q2·Straightforward
Point B(−2,5)B(-2, 5) is translated by the vector (3−4)\begin{pmatrix} 3 \\ -4 \end{pmatrix}. State the coordinates of B′B'.
Q3·Straightforward
Point C(0,−3)C(0, -3) is translated by the vector (−26)\begin{pmatrix} -2 \\ 6 \end{pmatrix}. State the coordinates of C′C'.
Q4·Straightforward
A rectangle has vertex D(5,−1)D(5, -1). It is translated by the vector (−3−2)\begin{pmatrix} -3 \\ -2 \end{pmatrix}. State the coordinates of D′D'.
Q5·Moderate
Point E(3,4)E(3, 4) is dilated from the origin by scale factor 22. State the coordinates of E′E'.
Q6·Moderate
Point F(−2,5)F(-2, 5) is dilated from the origin by scale factor 33. State the coordinates of F′F'.
Q7·Moderate
Triangle has vertex G(6,−4)G(6, -4). It is dilated from the origin by scale factor 12\dfrac{1}{2}. State the coordinates of G′G'.
Q8·Moderate
Point H(−4,6)H(-4, 6) is dilated from the origin by scale factor 32\dfrac{3}{2}. State the coordinates of H′H'.
Q9·Challenging
Point P(3,5)P(3, 5) is mapped to P′(6,10)P'(6, 10) by a dilation from the origin. What is the scale factor?
Q10·Challenging
Point Q(4,8)Q(4, 8) is mapped to Q′(1,2)Q'(1, 2) by a dilation from the origin. What is the scale factor?
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Q11·Challenging
Point R(−6,9)R(-6, 9) is mapped to R′(−2,3)R'(-2, 3) by a dilation from the origin. What is the scale factor?
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Q12·Challenging
Triangle ABCABC has side lengths 33 cm, 44 cm and 55 cm. It is mapped to triangle A′B′C′A'B'C' with side lengths 66 cm, 88 cm and 1010 cm. Classify this transformation.