NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Enlargement transformations

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A community garden in Canberra is designed with a rectangular plot measuring 51 m in length and 15 m in width. The gardeners decide to create an enlarged version of the plot using a scale factor of 4/3 from one of its corners.

What is the exact area of the enlarged rectangular plot in square metres? Express your answer as an exact fraction or whole number.

Concept illustrations

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Corresponding lengths scale together. The right triangles have side lengths 3,4,5 and 9,12,15. The length scale factor is 3 and the area scale factor is 9. Both use the same drawing scale.34912155×3
The right triangles have side lengths 3,4,5 and 9,12,15. The length scale factor is 3 and the area scale factor is 9. Both use the same drawing scale.
Coordinates on a plane. Illustration: the point (2, 2) lies on y = x. Read x across, then y up.-2-2-1-111223344xy
Illustration: the point (2, 2) lies on y = x. Read x across, then y up.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The original rectangular plot has a length of 51 m and a width of 15 m. The scale factor for enlargement is 4/3, applied from one corner, so both dimensions are scaled by this factor.
  2. To find the new length, multiply the original length by the scale factor: new_length = 51 × (4/3).
  3. Calculate 51 × (4/3): first divide 51 by 3 to get 17, then multiply by 4: 17 × 4 = 68. So, new_length = 68 m.
  4. To find the new width, multiply the original width by the scale factor: new_width = 15 × (4/3).
  5. Calculate 15 × (4/3): divide 15 by 3 to get 5, then multiply by 4: 5 × 4 = 20. So, new_width = 20 m.
  6. The area of a rectangle is given by length × width. Use the new dimensions: area = 68 × 20.
  7. Compute 68 × 20: 68 × 2 = 136, then multiply by 10: 136 × 10 = 1360.
  8. The area of the enlarged plot is 1360 square metres. This is an exact value, not a decimal approximation.
  9. Verify using the area scaling rule: area scales by the square of the scale factor. Original area = 51 × 15 = 765 m². Scale factor squared = (4/3)² = 16/9. New area = 765 × (16/9).
  10. Compute 765 × (16/9): divide 765 by 9 to get 85, then multiply by 16: 85 × 16 = 1360. Confirmed.
  11. The exact area of the enlarged rectangular plot is 1360 square metres.
  12. The result is consistent across two methods: direct scaling of dimensions and area scaling by square of scale factor.

Reference answer: 1360

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