Similarity and enlargement
Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids
What a learner can do afterwards
- Enlarge a shape by a given scale factor from a specified centre of enlargement
- Determine the scale factor between two similar shapes by comparing corresponding sides
- Explain why corresponding angles in similar shapes are equal while sides are in proportion
- Use the square of a scale factor to find how the area of a similar shape changes
1 · Read
Enlargement is a transformation that makes a shape bigger or smaller without changing its shape, like a photo blown up into a poster. You need two things: a centre of enlargement, which is a fixed point, and a scale factor, which tells you how many times bigger or smaller each length becomes.
To enlarge from the origin, multiply every coordinate by the scale factor. Triangle ABC has corners at (1, 1), (3, 1), and (1, 4). Enlarged by scale factor 2 from the origin, the new corners are (2, 2), (6, 2), and (2, 8).
When the centre is not the origin, you cannot just multiply the coordinates. Instead, work out how far each corner is from the centre, multiply that distance by the scale factor, then measure the same distance out from the centre in the same direction.
To go the other way, compare a pair of corresponding sides. Divide the length on the bigger shape by the matching length on the smaller one, and the answer is the scale factor. A side of 4 units matched with 12 units gives 12 ÷ 4 = 3, so every length has tripled.
Corresponding angles never change in an enlargement. Only the side lengths change, and every side scales by the same factor. That constant ratio is what makes the two shapes similar.
Area scales differently from length. If every length is multiplied by k, the area is multiplied by k times k, or k squared. A rectangle that is enlarged by 3 has 9 times the area because 3 squared is 9.
Enlargement keeps every angle the same while every length grows or shrinks by one constant scale factor measured from the centre.
The rewritten reading was written for this dataset. 'Similarity FAQ' is based on Khan Academy material, with its source identifier recorded per reading. 'Finding the scale factor' is original writing for this dataset. The area items retain their problem source credits and use clear wording for the area result.
2 · Watch
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