Transformations: Translation and Enlargement
A translation slides every point of a shape the same distance in the same direction, described by a column vector (x, y), where x is the horizontal shift and y is the vertical shift. An enlargement changes the size of a shape by a scale factor from a fixed centre of enlargement, multiplying every length and every distance from the centre by that factor, so a scale factor greater than 1 makes the shape bigger and one between 0 and 1 makes it smaller.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a translation, apply the column vector to every vertex: add the top number to each x-coordinate and the bottom number to each y-coordinate.
- A translation does not change the size, shape or orientation of the object; only its position changes.
- For an enlargement, identify the centre of enlargement and the scale factor.
- Find the distance from the centre to each vertex, multiply this distance by the scale factor, and plot the new vertex the same direction from the centre.
- Alternatively, use the rule: image coordinate = centre + scale factor x (object coordinate - centre), applied separately to the x- and y-coordinates.
- A scale factor bigger than 1 enlarges the shape; a scale factor between 0 and 1 makes the image smaller than the object, but it is still called an enlargement.
- Check your answer by confirming every side of the image is the scale factor times the matching side of the object.
Worked example
Triangle X has vertices A(2, 1), B(4, 1) and C(2, 3). Enlarge triangle X by scale factor 3, centre (0, 0), to give triangle X1. State the coordinates of the image vertices.
- Since the centre of enlargement is the origin (0, 0), multiply each coordinate of every vertex by the scale factor, 3.
- A(2, 1) maps to (2x3, 1x3) = (6, 3).
- B(4, 1) maps to (4x3, 1x3) = (12, 3).
- C(2, 3) maps to (2x3, 3x3) = (6, 9), so triangle X1 has vertices (6, 3), (12, 3) and (6, 9).
Practice questions
Try each question, then tap to reveal the answer.
Q1Translate the point (3, 5) by the column vector (4, -2), meaning 4 right and 2 down. State the image coordinates.Show answer
Answer: (3+4, 5-2) = (7, 3)
Q2Translate the point (-1, 6) by the column vector (-3, 2), meaning 3 left and 2 up. State the image coordinates.Show answer
Answer: (-1-3, 6+2) = (-4, 8)
Q3Point G is at (2, 3). After a translation, its image is at (9, 1). State the column vector of the translation.Show answer
Answer: (9-2, 1-3) = column vector (7, -2), meaning 7 right and 2 down
Q4Enlarge the point (3, 2) by scale factor 4, centre the origin. State the image coordinates.Show answer
Answer: (3x4, 2x4) = (12, 8)
Q5Enlarge the point (8, 12) by scale factor 1/2, centre the origin. State the image coordinates.Show answer
Answer: (8x0.5, 12x0.5) = (4, 6)
Q6A line segment has length 5 cm. It is enlarged by scale factor 6. Find the length of the image.Show answer
Answer: 5 x 6 = 30 cm
Q7A shape is enlarged by scale factor 2.5, centre (1, 1). One vertex of the object is at (5, 1). Find the coordinates of its image.Show answer
Answer: x: 1 + 2.5x(5-1) = 11; y: 1 + 2.5x(1-1) = 1, so the image is (11, 1)
Q8A rectangle has an area of 12 cm^2. It is enlarged by scale factor 3. A student says the area of the image is 36 cm^2. Explain why the student is wrong, and find the correct area.Show answer
Answer: The student only multiplied by the scale factor once, but area scales by the scale factor squared; correct area = 12 x 3^2 = 12 x 9 = 108 cm^2
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Shape M is translated by the column vector (-5, 3), meaning 5 left and 3 up, to give shape M1. (a) A vertex of shape M is at (6, -2). Find the coordinates of the image of this vertex in shape M1. (b) State the column vector that would translate shape M1 back to shape M.
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Rectangle R has vertices (2, 1), (6, 1), (6, 3) and (2, 3). Rectangle R is enlarged by scale factor 2, centre (2, 1), to give rectangle R1. (a) Find the coordinates of all four vertices of R1. (b) Find the area of rectangle R1, and state the ratio of the area of R to the area of R1.
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Free printable worksheet
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This topic is chapter 17 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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