Enlargement About (h,k)

\( (x,y)\mapsto (\,h+k(x-h),\ k(y-k)+k\,) \)
Transformations GCSE
Question 7 of 20
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Enlarge (4,2) about (1,1) with scale factor 2.

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Subtract centre, multiply, then add back.

Explanation

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Statement

An enlargement about a point changes the size of a shape while keeping the same proportions, with the enlargement centred at a fixed point \((h,k)\). The transformation is given by:

\[ (x,y) \;\mapsto\; \big(h + k(x-h),\; k(y-k)+k\big) \]

Here, \((h,k)\) is the centre of enlargement and \(k\) is the scale factor.

Why it’s true

  • Enlargement stretches or shrinks each point relative to the centre \((h,k)\).
  • The distance from the centre to each point is multiplied by the scale factor \(k\).
  • If \(k > 1\), the image is an enlargement; if \(0 < k < 1\), the image is a reduction; if \(k < 0\), the image is enlarged and reflected through the centre.

Recipe (how to use it)

  1. Write down the coordinates of the point to be transformed, \((x,y)\).
  2. Subtract the centre of enlargement coordinates \((h,k)\).
  3. Multiply the differences by the scale factor \(k\).
  4. Add back the centre coordinates \((h,k)\) to find the new point.

Spotting it

Enlargement questions mention a “centre of enlargement” and a “scale factor”.

Common pairings

  • Other transformations (reflection, rotation, translation).
  • Similarity in triangles (enlargements preserve shape and angles).
  • Coordinate geometry when working with transformations.

Mini examples

  1. Given: Centre \((0,0)\), scale factor 2, point \((3,4)\). Answer: \((6,8)\).
  2. Given: Centre \((1,1)\), scale factor 3, point \((2,4)\). Answer: \((4,10)\).

Pitfalls

  • Forgetting to subtract the centre before scaling.
  • Misinterpreting negative scale factors (they include a reflection).
  • Mixing enlargement with translation (enlargement changes distances, not just positions).

Exam strategy

  • Plot the centre clearly on your diagram before enlarging.
  • Work step by step: subtract → multiply → add back.
  • Check by comparing relative distances after enlargement.

Summary

An enlargement about a point \((h,k)\) with scale factor \(k\) stretches/shrinks a shape relative to that centre. The formula ensures proportional scaling while preserving angles and the centre as a fixed point.

Worked examples

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  1. Enlarge (2,3) about (0,0) with scale factor 2.
    1. Subtract centre: (2,3)
    2. Multiply by 2: (4,6)
    3. Add back centre (0,0): (4,6)
    Answer: (4,6)
  2. Enlarge (5,7) about (0,0) with scale factor 3.
    1. Subtract centre: (5,7)
    2. Multiply by 3: (15,21)
    3. Add back centre (0,0): (15,21)
    Answer: (15,21)
  3. Enlarge (4,1) about (1,1) with scale factor 2.
    1. \( Subtract centre: (4-1,1-1)=(3,0) \)
    2. Multiply by 2: (6,0)
    3. Add back centre (1,1): (7,1)
    Answer: (7,1)
  4. Enlarge (3,6) about (2,2) with scale factor 3.
    1. Subtract centre: (1,4)
    2. Multiply by 3: (3,12)
    3. Add back centre: (5,14)
    Answer: (5,14)
  5. Enlarge (0,5) about (0,0) with scale factor 0.5.
    1. Subtract centre: (0,5)
    2. Multiply by 0.5: (0,2.5)
    3. Add back centre: (0,2.5)
    Answer: (0,2.5)
  6. Enlarge (6,2) about (1,1) with scale factor -2.
    1. Subtract centre: (5,1)
    2. Multiply by -2: (-10,-2)
    3. Add back (1,1): (-9,-1)
    Answer: (-9,-1)
  7. Enlarge (10,4) about (2,2) with scale factor 0.5.
    1. Subtract centre: (8,2)
    2. Multiply by 0.5: (4,1)
    3. Add back (2,2): (6,3)
    Answer: (6,3)
  8. Enlarge (8,5) about (3,2) with scale factor 4.
    1. Subtract centre: (5,3)
    2. Multiply by 4: (20,12)
    3. Add back (3,2): (23,14)
    Answer: (23,14)
  9. Enlarge (-2,7) about (1,1) with scale factor -1.
    1. Subtract centre: (-3,6)
    2. Multiply by -1: (3,-6)
    3. Add back (1,1): (4,-5)
    Answer: (4,-5)
  10. Enlarge (12,8) about (0,0) with scale factor 1/4.
    1. Subtract centre: (12,8)
    2. Multiply by 0.25: (3,2)
    3. Add back (0,0): (3,2)
    Answer: (3,2)