Negative Scale Factor Enlargement [H]

Enlargement with Negative Scale Factors

A negative scale factor combines an enlargement with a rotation of 180° about the centre of enlargement. The image appears on the opposite side of the centre from the original, and is inverted (turned upside down).

How It Works

For scale factor \( k < 0 \), the construction rule is the same as for positive enlargement, except the image point is placed on the opposite ray through the centre:

  1. Draw a line from the centre of enlargement through each vertex of the original shape.
  2. Extend the line beyond the centre (on the other side).
  3. Measure \( |k| \) times the distance from the centre along this extended ray — the image vertex lies there.

The image is similar to the original: all lengths are multiplied by \( |k| \), all angles are preserved, but the orientation is reversed.

C A B D A' B' D' Enlargement, scale factor \(k=-1.5\), centre C

Effect on Coordinates

If the centre of enlargement is the origin and the scale factor is \( k \), each point \( (x, y) \) maps to \( (kx, ky) \). For \( k = -2 \):

\[ (3, 1) \xrightarrow{k=-2,\ \text{centre } (0,0)} (-6, -2) \]

If the centre is not the origin, subtract the centre coordinates first, multiply by \( k \), then add the centre back.

Worked Example

Triangle with vertices \( A(1,3) \), \( B(3,3) \), \( D(1,5) \). Enlarge by scale factor \( -2 \) from centre \( C(0,1) \).

For each vertex, compute: image = centre + \( k \times \)(vertex − centre).

\[ A' = (0,1) + (-2)\times((1,3)-(0,1)) = (0,1) + (-2)(1,2) = (0,1)+(-2,-4) = (-2,-3) \] \[ B' = (0,1) + (-2)(3,2) = (0,1)+(-6,-4) = (-6,-3) \] \[ D' = (0,1) + (-2)(1,4) = (0,1)+(-2,-8) = (-2,-7) \]

The image is inverted and on the opposite side of \( C \).

 Key Takeaways

  • Negative scale factor = enlargement by \( |k| \) plus 180° rotation about the centre.
  • The image is on the opposite side of the centre, inverted relative to the original.
  • Construction: extend rays beyond the centre; measure \( |k| \times \) distance on the far side.
  • Coordinate rule: image = centre + \( k \times \)(vertex − centre). Works for any centre.
  • The image is similar (not congruent) to the original — all lengths scaled by \( |k| \).