Congruent and similar shapes; enlargements with fractional and negative scale factors
Congruent shapes have the same size and shape. Similar shapes have the same shape but different size — corresponding lengths are in a fixed ratio (the scale factor).
Enlargement transformations
An enlargement is defined by:
- A centre of enlargement (a point).
- A scale factor (a number).
Every point P maps to P' such that the centre, P and P' are collinear, with O P' = SF × OP.
Positive scale factor > 1
The image is larger and on the same side of the centre as the object.
Positive scale factor < 1 (fractional)
The image is smaller and on the same side of the centre. SF = 1/2 means image is half the size.
Worked example: a triangle with vertices A(2, 1), B(4, 1), C(2, 4) is enlarged by SF 1/2 from origin.
- A' = (1, 0.5), B' = (2, 0.5), C' = (1, 2).
Negative scale factor
The image is on the opposite side of the centre, and is inverted (rotated 180°). |SF| determines the size.
Worked example: square with vertices (1,1), (3,1), (3,3), (1,3) enlarged by SF −2 from origin.
- (1,1) → (−2,−2); (3,1) → (−6,−2); (3,3) → (−6,−6); (1,3) → (−2,−6).
- The image is twice as large and on the opposite side of the origin.
Constructing an enlargement
- From centre O, draw rays through each vertex of the object.
- Measure OP. Multiply by SF (sign included). Mark P' along the ray (or backwards if negative SF).
- Repeat for each vertex; join up the image.
Identifying the centre and SF from a diagram
To find:
- Centre: extend lines connecting corresponding object/image vertices — they all pass through the centre.
- SF: ratio of any pair of corresponding lengths (image/object).
⚠Common mistakes
- Confusing fractional with negative scale factor — fractional shrinks; negative flips.
- Forgetting to flip for negative scale factor — image is on the OTHER side of centre.
- Measuring length only, not the directed distance from centre — for enlargement, position matters.
- Using "scale factor" loosely — must give value AND centre to define the enlargement.
- Treating an enlargement as similar but ignoring orientation — congruence/similarity ignore orientation; transformations include orientation info.
➜Try this— Quick check
A point P(4, 2) is enlarged by SF −1.5 about the origin. Find P'.
- P' = −1.5 × (4, 2) = (−6, −3).
AI-generated · claude-opus-4-7 · v3-deep-geometry