One-page revision sheet

• Coordinates are written (x, y) — across first, then up. (3, 1) and (1, 3) are different points.
• The origin is (0, 0), where the axes cross.
• The four quadrants are numbered anticlockwise from the top right: 1 (+, +), 2 (−, +), 3 (−, −), 4 (+, −).
• A point on an axis is in no quadrant.
• If two points share a y-coordinate the line is horizontal: subtract the x-coordinates.
• If two points share an x-coordinate the line is vertical: subtract the y-coordinates.
• A distance is never negative — take the size of the subtraction.
• The midpoint of such a line is the average of the two coordinates.
• None of this needs a grid; it is arithmetic on the coordinates.

Key words: coordinates  ·  origin  ·  quadrant  ·  x-coordinate  ·  y-coordinate  ·  midpoint  ·  axis

• A scale tells you what one length on a drawing stands for in real life.
• Drawing → real: multiply. Real → drawing: divide.
• A scale can be written as “1 cm represents 5 m” or as the ratio 1 : 500.
• For a ratio scale, both sides must be in the same units: 5 m = 500 cm, so 1 cm to 5 m is 1 : 500.
• The bigger the second number, the smaller the drawing.
• Useful conversions: 100 cm = 1 m, 100 000 cm = 1 km.
• Every angle is unchanged, and every length is multiplied by the same number — the shape does not distort.
• Areas do not scale by the same number: doubling every length multiplies the area by 4.

Key words: scale  ·  scale drawing  ·  ratio scale  ·  plan  ·  map  ·  represents

• A translation slides a shape. Nothing turns and nothing flips.
• It is described by a vector (across, up).
• To translate a point, add the vector to its coordinates.
• To find the vector, do image − object.
• A negative first number means left; a negative second number means down.
• Two translations one after the other are the same as one translation by the sum of the vectors.
• The image is congruent to the object: same lengths, same angles, same way round.
• To undo a translation, reverse the signs of both parts.
• None of this needs a grid — the arithmetic on the coordinates is the whole method.

Key words: translation  ·  vector  ·  object  ·  image  ·  congruent  ·  corresponding points

• A reflection flips a shape in a mirror line.
• Every point and its image are the same distance from the mirror, on opposite sides, measured perpendicular to it.
• Reflect in the x-axis: (x, y) → (x, −y).
• Reflect in the y-axis: (x, y) → (−x, y).
• Reflect in y = x: (x, y) → (y, x) — the coordinates swap.
• A point on the mirror line does not move. It is invariant.
• The image is congruent: every length and every angle is unchanged.
• But the shape is turned over, so the vertices run the other way round.
• To find an unknown mirror line, join a point to its image; the mirror cuts that join in half at right angles.

Key words: reflection  ·  mirror line  ·  image  ·  congruent  ·  invariant point  ·  perpendicular

• A rotation turns a shape about a centre of rotation.
• To describe one fully you need three things: the angle, the direction and the centre.
• About the origin, 90° anticlockwise: (x, y) → (−y, x).
• About the origin, 180°: (x, y) → (−x, −y).
• About the origin, 90° clockwise: (x, y) → (y, −x).
• For 180° the direction makes no difference.
• The centre does not move. Everything else does.
• About any other centre, work with the offset from the centre, turn that, then add the centre back on.
• The image is congruent, and unlike a reflection the vertices still run the same way round.

Key words: rotation  ·  centre of rotation  ·  anticlockwise  ·  clockwise  ·  quarter turn  ·  half turn

• An enlargement multiplies every length by a scale factor.
• It needs two things: the scale factor and the centre of enlargement.
• From the origin: (x, y) → (kx, ky).
• From any other centre: multiply the offset from the centre by k, then add the centre back on.
• Angles do not change. Only lengths do.
• The image is mathematically similar to the object — same shape, different size. It is not congruent unless k = 1.
• Lengths and perimeters are multiplied by k.
• Areas are multiplied by k², because an area is a length times a length.
• The centre of enlargement is the one point that does not move.
• A scale drawing is an enlargement — section 14.2 was doing this all along.

Key words: enlargement  ·  scale factor  ·  centre of enlargement  ·  similar  ·  congruent  ·  ratio