Mr Kidman's Classroom
Revise the key ideas
Describing transformations
- A translation moves every point by the same vector (a movement with a size and direction). The vector (3, −2) means 3 units right and 2 down. In column-vector notation, write the horizontal component 3 above the vertical component −2. A complete description needs both movements: horizontal and vertical.
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-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 rotation - An enlargement needs a centre and a scale factor. Draw rays (straight lines extending outwards) 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 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.
SSS means three equal matching sides; SAS means two equal sides and their included angle; ASA/AAS means two equal angles and a matching side. RHS means right angle, 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 (its size without the minus sign); 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 several labelled vertices and the way the shape is positioned to identify the resulting transformation.
Translation
- Fixed points — A point on a reflection's mirror line stays in its original position: reflecting (2, 3) in x = 2 leaves it at (2, 3). Such a point is invariant. Under a rotation through a non-zero angle less than 360°, the centre is fixed, while other points move. An enlargement with scale factor 2 fixes its centre but doubles distances of other points from that centre. Distinguish an unchanged position from an unchanged length or angle; these are different properties of a transformation.
