GCSE Maths · Transformations and Similarity · Higher · Summary

GCSE Maths Transformations and Similarity (Higher): summary and key words

Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Reflection	Mirror line, same distance on the other side. Describe with the equation of the mirror line. y = x swaps the coordinates; y = −x swaps and changes both signs.
Rotation	Angle + direction + centre. Write each vertex as a move from the centre and turn it: 90° clockwise (a, b) → (b, −a); 90° anticlockwise (−b, a); 180° (−a, −b).
Translation	Column vector: top = across (right +), bottom = up/down (up +). Vector = image − object.
Enlargement	Scale factor + centre. Multiply the distance from the centre by the scale factor. A fraction makes the shape smaller; a negative scale factor puts the image on the other side of the centre (Higher).
Describe fully	Name the type and give its detail: line / angle, direction, centre / vector / scale factor, centre.
Combinations (Higher)	Do them in order. Two reflections in parallel lines = translation; in crossing lines = rotation. Invariant point: unchanged by the transformation.
Congruent triangles	SSS, SAS (included angle), ASA / AAS, RHS. AAA and SSA are not enough.
Similar shapes	Equal angles, sides in the same ratio. Scale factor = large ÷ small. Small to large: multiply. Large to small: divide.
Area and volume	Length scale factor k: area × k², volume × k³. Ratio a : b gives a² : b² and a³ : b³. Take roots to go back.

I can reflect a shape in x = a, y = b, y = x and y = −x.
I can rotate a shape about any centre through 90° or 180° and describe a rotation fully.
I can translate a shape using a column vector and find the vector of a translation.
I can enlarge a shape with a whole-number or fractional scale factor from any centre, and describe an enlargement fully.
I can describe fully any single transformation, and I know which are congruent and which are similar.
Higher: I can use negative scale factors, combine transformations and find invariant points.
I can decide whether two triangles are congruent using SSS, SAS, ASA or RHS and give a reason.
I can find a missing length in similar shapes using a scale factor, including similar triangles from parallel lines.
I can use k, k² and k³ to compare lengths, areas and volumes of similar shapes.
Higher: I can prove that two triangles are congruent or similar.

Transformation: A change to a shape: reflection, rotation, translation or enlargement.
Object and image: The object is the original shape; the image is the shape after the transformation.
Mirror line: The line a shape is reflected in. Each point and its image are the same distance from it.
Centre of rotation: The fixed point a shape is turned about.
Column vector: Two numbers written one above the other. The top number is the move across and the bottom number is the move up or down.
Scale factor: The number every length is multiplied by in an enlargement or between similar shapes.
Centre of enlargement: The fixed point that distances are measured from in an enlargement.
Invariant point: A point that does not move when a transformation is carried out.
Congruent: Exactly the same size and shape (one can be placed on top of the other).
Similar: The same shape but not necessarily the same size: equal angles and sides in the same ratio.
Corresponding: In the same position in two shapes (sides or angles that match).
Included angle: The angle between two given sides.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.