Mathematics (Core)Exam code: 0580

Transformations

Transformations

PractiseFlashcards

Doing one

  • An enlargement resizes the shape by a scale factor from a fixed centre of enlargement
  • Each vertex moves along the line from the centre through it, ending scale-factor times as far from the centre
  • Multiply each distance from the centre, both across and up, by the scale factor
Shape C enlarged to C prime with the centre of enlargement marked at (2, 1): a red ray is drawn from the centre through a vertex of C and on to the matching vertex of C prime, which sits twice as far along it
Source: Enlargements by Save My Exams
  • A scale factor between 0 and 1 makes the image smaller, even though it is still called an enlargement
A large shape enlarged by a fraction to a much smaller green image: the rays from the centre of enlargement at (4, 2) run from each vertex of the object inwards, and the image sits between the object and the centre
Source: Enlargements by Save My Exams
  • Every length multiplies by the scale factor, so the image is similar to the object rather than congruent

Describing one

  • Say enlargement, then give the scale factor, then the centre as coordinates
  • Find the centre by drawing a line through each pair of corresponding vertices and seeing where they all cross
Working backwards to find a centre of enlargement: a red line is ruled through three pairs of matching vertices of the large shape A and the small shape B, and all of the lines meet at one point, which is the centre
Source: Enlargements by Save My Exams
  • Find the scale factor by dividing an image length by the matching object length

Worked example

Describing an enlargement

Triangle A has vertices (1, 1), (3, 1) and (1, 2). Triangle B has vertices (2, 2), (6, 2) and (2, 4). Describe fully the single transformation that maps A onto B.

A coordinate grid with x from 0 to 7 and y from 0 to 5. Triangle A is drawn with vertices at (1, 1), (3, 1) and (1, 2), and the larger triangle B with vertices at (2, 2), (6, 2) and (2, 4). No centre of enlargement and no rays are drawn.

Solution:

  • B is bigger than A, so the only candidate is an enlargement: the other three keep the size the same
  • Compare a pair of matching sides: A has a base from (1, 1) to (3, 1), which is 2 units, and B has a base from (2, 2) to (6, 2), which is 4 units
  • Scale factor = 4 ÷ 2 = 2
  • Draw a line through (1, 1) and (2, 2), another through (3, 1) and (6, 2), and a third through (1, 2) and (2, 4)
  • All three meet at the origin, so that is the centre
  • The full description is an enlargement, scale factor 2, centre (0, 0)
  • Check: each vertex of B is exactly twice as far from the origin as the matching vertex of A

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