Mr Kidman's Classroom← Back

Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M29 · Transformations and congruence

Revision notes, worked examples and methods for transformations and congruence.

Revision notes ready · Quizzes and videos coming soon.

Revise the key ideas

Describing transformations

  • A translation moves every point by the same vector. (3−2) means 3 units right and 2 down. A complete description needs both components.
    A triangle translated three right and two downEvery vertex moves three horizontal units right and two vertical units down, preserving shape and size.01234567-10123456xy3 right, 2 down
    A triangle translated three right and two down
    Enlarge diagram: A triangle translated three right and two down
  • A reflection needs its mirror line, for example x = 2 or y = x. Perpendicular distances from corresponding points to the mirror line are equal.
  • Reflecting (a, b) in the x-axis gives (a, −b); in the y-axis gives (−a, b); in y = x gives (b, a).
  • Worked example: A triangle with vertices (1, 1), (3, 1), (1, 3) reflects in the y-axis to (−1, 1), (−3, 1), (−1, 3). Each corresponding point is equally far from the mirror line.
    Reflection of a triangle in the y-axisThe original triangle at one one, three one and one three reflects to minus one one, minus three one and minus one three.-4-2024-101234xyimageoriginal
    Reflection of a triangle in the y-axis
    Enlarge diagram: Reflection of a triangle in the y-axis
  • A rotation needs centre, angle and direction unless the angle is 180°. A 90° anticlockwise rotation about the origin sends (a, b) to (−b, a).
  • Worked example: Rotating (3, 1) through 90° anticlockwise about (0, 0) gives (−1, 3). Draw centre-to-point radii to check that the distance from the centre stays unchanged.
    A ninety-degree anticlockwise rotationThe vector from the origin to three one rotates ninety degrees anticlockwise to minus one three. Both vectors have equal length and the same coordinate scale.(3, 1)(−1, 3)O90°Anticlockwise rotation about the origin
    A ninety-degree anticlockwise rotation
    Enlarge diagram: A ninety-degree anticlockwise rotation
  • An enlargement needs a centre and a scale factor. Draw rays from the centre through each vertex and multiply each distance from the centre by the scale factor.
  • Worked example: Enlarging (3, 1) by scale factor 2 about (1, 1) doubles the relative vector (2, 0), giving (5, 1). Doubling the original coordinates works only for a centre at the origin.
  • Join each vertex and its image to the enlargement centre; corresponding points must lie on those rays. For scale factor 2, an original vertex at distance d moves to distance 2d from the centre.
    An enlargement with scale factor twoThe image triangle is twice the original distances from the marked centre. All corresponding vertices lie on common rays.centreEach centre-to-vertex vector doubles
    An enlargement with scale factor two
    Enlarge diagram: An enlargement with scale factor two
  • A positive scale factor below 1 reduces the shape. Enlargement by halves every length but keeps all angles equal.
  • Reflections, rotations and translations preserve lengths and angles, so images are congruent. Enlargement preserves angles and proportions; it generally changes size.

Triangle congruence

  • Congruent triangles have corresponding sides and angles equal. Valid tests are SSS, SAS, ASA (or AAS) and RHS for right-angled triangles with equal hypotenuse and one other side.
  • In SAS, the equal angle must be included between the two equal sides. SSA does not generally prove congruence; AAA proves similarity only.
  • When using congruence in a proof, identify corresponding vertices in order, name the test, then use the corresponding equal lengths or angles.

Higher — negative factors and combinations

  • A negative enlargement places the image on the opposite side of the centre. Distances are multiplied by the magnitude of the factor; a factor −2 also reverses the direction of each centre-to-point vector.
  • Worked example: Enlarging (3, 1) by factor −2 about (1, 1) gives (−3, 1), because (1, 1) + (−2)(2, 0) = (−3, 1).
  • For combined transformations, apply them in the stated order; changing the order can change the result. Two translations combine by adding their vectors.
  • Two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation. Track a point and orientation to identify the resulting transformation.

Test yourself

Quiz coming soon

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.

Revision video

Video coming soon.