Enlarging with Negative Scale Factors
Enlarging with a negative scale factor produces an image on the opposite side of the centre of enlargement to the object, rotated 180 degrees compared with the original shape. The size of the image is found using the magnitude of the scale factor, so a scale factor of -3 makes an image 3 times the size of the object. This Higher GCSE topic extends ordinary enlargement to negative scale factors.
Before you start
Make sure you're comfortable with these topics first:
Method
- Find the vector from the centre of enlargement to each vertex of the object (the difference in x and y coordinates).
- Multiply that vector by the scale factor, including its negative sign.
- Add the result to the coordinates of the centre of enlargement to find each image vertex.
- Remember the image appears on the opposite side of the centre from the object, and upside down (rotated 180 degrees) compared with it.
- Use the magnitude (ignore the sign) of the scale factor for any length, area or volume calculation.
- To find an area scale factor, square the magnitude of the scale factor, since area scale factor is always positive.
Worked example
Triangle ABC has vertices A(4, 1), B(6, 1) and C(4, 3). It is enlarged by scale factor -2, centre (0, 0), to give triangle A'B'C'. Find the coordinates of A', B' and C'.
- Find the vector from the centre (0, 0) to each vertex: A is (4, 1), B is (6, 1), C is (4, 3).
- Multiply each vector by the scale factor -2: A becomes (-8, -2), B becomes (-12, -2), C becomes (-8, -6).
- Since the centre is the origin, these are already the image coordinates.
- A' = (-8, -2), B' = (-12, -2), C' = (-8, -6).
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Give two facts describing how the image compares with the original shape when an enlargement uses a negative scale factor.
Triangle DEF has vertices D(1, 1), E(4, 1) and F(1, 3). Triangle DEF is enlarged by scale factor -3, centre (0, 0), to give triangle D'E'F'. Find the coordinates of D', E' and F'.
Triangle PQR has vertices P(2, 2), Q(4, 2) and R(2, 3). It is enlarged to form triangle P'Q'R' with vertices P'(-2, -2), Q'(-8, -2) and R'(-2, -5). Find the scale factor of the enlargement and the coordinates of the centre of enlargement.
Free printable worksheet
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