What is Enlargement?

Enlargement in geometry is a transformation that changes the size of a figure, but not its shape. This means that the transformed figure is similar to the original, maintaining the same proportions and angles.

Key Concepts of Enlargement

Scale Factor

The scale factor is the ratio by which the figure is enlarged or reduced. If the scale factor is greater than 1, the figure enlarges. If it is between 0 and 1, the figure reduces in size. For example, a scale factor of 2 will double the size of the figure, while a scale factor of 0.5 will halve it.

Center of Enlargement

The center of enlargement is the fixed point from which the figure is enlarged or reduced. Every point on the original figure moves away from or towards this center, depending on whether the scale factor is greater or less than 1.

How to Perform an Enlargement

  1. Identify the Center of Enlargement: Choose a point that will act as the center.
  2. Determine the Scale Factor: Decide how much you want to enlarge or reduce the figure.
  3. Draw Lines from the Center: Draw lines from the center of enlargement to each vertex of the original figure.
  4. Apply the Scale Factor: Measure the distance from the center to each vertex and multiply it by the scale factor. Mark the new points.
  5. Connect the Points: Connect the new points to form the enlarged or reduced figure.

Example

Imagine you have a triangle with vertices at points A, B, and C. If the center of enlargement is point O and the scale factor is 2, you would draw lines from O to A, B, and C. Measure the distances OA, OB, and OC, and then double them. Mark the new points as A’, B’, and C’, and connect them to form the enlarged triangle.

Mathematical Representation

If $(x, y)$ are the coordinates of a point on the original figure, and $(x’, y’)$ are the coordinates of the corresponding point on the enlarged figure, the transformation can be represented as:

$x’ = kx$

$y’ = ky$

where $k$ is the scale factor.

Practical Applications

Enlargement is used in various fields, such as architecture, art, and engineering. For instance, blueprints of buildings are often enlarged to create detailed models. Similarly, artists might use enlargement techniques to create murals from smaller sketches.

Conclusion

Understanding enlargement is crucial for grasping how shapes and figures can be transformed while maintaining their proportions. It has wide-ranging applications, making it an essential concept in geometry and beyond.

3. BBC Bitesize – Transformations

Citations

  1. 1. Khan Academy – Geometric Transformations
  2. 2. Math is Fun – Enlargement

Related

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H + HO2 → O2 + H2 k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O2 k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) H + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-5 s^-1) φ

Table 1 Reactions, rate constants and activation energies used in the model* No. Reaction kopt (M⁻¹ s⁻¹) 1 OH + H₂ → H + H₂O 3.74 x 10⁷ 2 OH + HO₂ → HO₂ + OH⁻ 5 x 10⁹ 3 OH + H₂O₂ → HO₂ + H₂O 3.8 x 10⁷ 4 OH + O₂ → O₂ + OH 9.96 x 10⁹ 5 OH + HO₂ → O₂ + H₂O 7.1 x 10⁹ 6 OH + OH → H₂O₂ 5.3 x 10⁹ 7 OH + e⁻aq → OH⁻ 3 x 10¹⁰ 8 H + O₂ → HO₂ 2.0 x 10¹⁰ 9 H + HO₂ → H₂O₂ 2.0 x 10¹⁰ 10 H + H₂O₂ → OH + H₂O 3.44 x 10⁷ 11 H + OH → H₂O 1.4 x 10¹⁰ 12 H + H → H₂ 1.94 x 10¹⁰ 13 e⁻aq + O₂ → O₂⁻ 1.9 x 10¹⁰ 14 e⁻aq + O₂ → HO₂⁻ + OH⁻ 1.3 x 10¹⁰ 15 e⁻aq + HO₂ 2.0 x 10¹⁰ 16 e⁻aq + H₂O₂ 1.1 x 10¹⁰ 17 e⁻aq + HO₂ → OH + OH⁻ 1.3 x 10¹⁰ 18 e⁻aq + H⁺ → H 2.3 x 10¹⁰ 19 e⁻aq + e⁻aq → H₂ + OH⁻ + OH⁻ 2.5 x 10⁹ 20 HO₂ + O₂ → O₂ + HO₂ 1.3 x 10⁹ 21 HO₂ + HO₂ → O₂ + H₂O₂ 8.3 x 10⁵ 22 HO₂ + HO₂ → O₂ + OH + H₂O 3.7 23 HO₂ + HO₂ → O₂ + O₂ + OH + H₂O 7 x 10⁵ s⁻¹ 24 H⁺ + O₂⁻ → HO₂ 4.5 x 10¹⁰ 25 H⁺ + O₂⁻ → O₂ 2.0 x 10¹⁰ 26 H⁺ + OH⁻ 1.4 x 10¹¹ 27 H⁺ + HO₂⁻ 2 x 10¹⁰ 28 H₂O₂ → HO₂ + H⁺ + OH⁻ 2.5 x 10⁻⁵ s⁻¹ 29 H₂O₂ → H⁺ + OH⁻ 1.4 x 10⁻⁷ s⁻¹ 30 O₂ + O₂ → O₂ + HO₂ + OH⁻ 0.3 31 O₂ + H₂O₂ → O₂ + OH + OH 16 32

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H2O + O → 2 OH k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) OH + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-8 s^-1) φ