Math 9 · Similarity and transformations
Same shape, different size
Two figures are similar when one is an exact enlargement or reduction of the other. One number, the scale factor, connects every pair of matching lengths, so knowing one pair tells you all of them. The unit ends with symmetry: what a shape does when it is folded or turned onto itself.
- 1. The words, first
- 2. Scale factor and scale diagrams
- 3. Similar polygons and similar triangles
- 4. Line and rotation symmetry
- 5. What costs marks
The words, first
The idea: Corresponding is the word that matters most. Every calculation here compares a length with the one in the same position.
| Word | What it means |
|---|---|
| Scale factor | The number every length is multiplied by: diagram length ÷ actual length. Written as a number (0.5), a fraction (1/200) or a ratio (1 : 200). |
| Scale diagram | A drawing that is an exact enlargement or reduction of a real object, such as a floor plan. |
| Enlargement / reduction | A scale factor greater than 1, or between 0 and 1. |
| Corresponding | In the same position in both figures. Corresponding angles match up, and so do corresponding sides. |
| Similar polygons | Polygons whose corresponding angles are equal and whose corresponding sides are all in the same ratio. |
| ~ | “Is similar to.” △ABC ~ △DEF says triangle ABC is similar to triangle DEF, with the vertices listed in matching order. |
| Line of symmetry | A line that splits a shape into two mirror-image halves. |
| Rotation symmetry | A shape has it if it fits onto itself during a turn of less than 360° about its centre. |
| Order and angle of rotation | How many times it fits onto itself in one full turn, and the smallest turn that does it: 360° ÷ the order. |
Scale factor and scale diagrams
The idea: Scale factor = diagram length ÷ actual length, with both in the same units. Greater than 1 is an enlargement; less than 1 is a reduction.
Finding it. A photo 3 cm wide is enlarged to 7.5 cm wide. The scale factor is 7.5 ÷ 3 = 2.5, so if the photo was 2 cm tall, the enlargement is 2 × 2.5 = 5 cm tall.
Units first. A floor plan says 1 cm represents 2 m. Change the metres to centimetres before calling it a scale factor: 2 m = 200 cm, so the scale is 1 : 200.
Worked example. On that plan a room is 4.5 cm long and 3.2 cm wide. How big is the real room?
4.5 cm × 200 = 900 cm = 9 m · 3.2 cm × 200 = 640 cm = 6.4 m
From plan to real life, multiply by 200; from real life to plan, divide. Real rooms are bigger than drawings of them, so the numbers should grow.
Similar polygons and similar triangles
The idea: Match up the corresponding sides first. Every pair then has the same ratio, and that ratio is the scale factor.
Polygons need both conditions. A square and a long rectangle have equal angles but are not similar — the rectangle is stretched. A square and a rhombus with no right angles have proportional sides but different angles, so they are not similar either.
Triangles are the exception. If two triangles have equal corresponding angles, their sides are automatically in the same ratio, and the other way round. Checking either one is enough.
Worked example. △ABC ~ △DEF, with AB = 6 cm, BC = 8 cm, AC = 9 cm and DE = 9 cm. Find EF and DF.
- Match the sides from the letter order: A with D, B with E, C with F. So AB goes with DE, BC with EF, and AC with DF.
- Scale factor from ABC to DEF: DE ÷ AB = 9 ÷ 6 = 1.5.
- Multiply: EF = 8 × 1.5 = 12 cm, and DF = 9 × 1.5 = 13.5 cm.
Measuring what you cannot reach. At the same moment, a 1.8 m student casts a 2.4 m shadow and a tree casts a 16 m shadow. The sun strikes both at the same angle, so the triangles are similar:
height ÷ 16 = 1.8 ÷ 2.4 = 0.75 → height = 16 × 0.75 = 12 m
Math 10C turns this same fact — the ratio of two sides depends only on the angles — into trigonometry.
Line and rotation symmetry
The idea: Line symmetry is a fold; rotation symmetry is a turn. Count the folds that match, and count the fits in one full turn.
| Shape | Lines of symmetry | Order of rotation | Angle of rotation |
|---|---|---|---|
| Square | 4 | 4 | 90° |
| Rectangle, not a square | 2 | 2 | 180° |
| Equilateral triangle | 3 | 3 | 120° |
| Regular hexagon | 6 | 6 | 60° |
| Parallelogram, not a rectangle or rhombus | 0 | 2 | 180° |
Count each fit once. A square fits at 90°, 180°, 270° and 360° — four times, so order 4. A shape that fits back only at 360° has order 1, which means no rotation symmetry.
Worked example. A logo has rotation symmetry of order 5. Its angle of rotation is 360° ÷ 5 = 72°, so it also fits at 144°, 216° and 288°.
Completing a shape. Reflect each vertex across the line of symmetry: the same distance away, straight across on the other side. With the line x = 2, the point (5, 3) is 3 units to the right of it, so its image is 3 units to the left: (−1, 3).
What costs marks
The idea: Direction and matching, mostly.
- Scale factor upside down. It is diagram ÷ actual, so a reduction must come out less than 1.
- Mixed units in a scale. 1 cm to 2 m is 1 : 200, not 1 : 2.
- Matching the wrong sides. Use the letter order, or match shortest with shortest.
- Calling polygons similar because their angles match. Only triangles get away with that.
- Giving a parallelogram two lines of symmetry. It has none: folding along a diagonal does not make the halves match.