The Impact of Multiplying Coordinates by a Negative Number

In geometry, understanding how transformations affect geometric figures is crucial. One such transformation is dilation, which involves changing the size of a figure while maintaining its shape. Multiplying the coordinates of a figure by a negative number is a specific type of dilation that can lead to interesting changes in its dimensions and properties.

Understanding Dilation

Dilation is a transformation that scales a geometric figure by a specific factor. This factor, known as the scale factor, determines how much the figure is enlarged or reduced. When the scale factor is greater than 1, the figure is enlarged; when it’s between 0 and 1, the figure is reduced. A negative scale factor introduces an additional element: reflection.

The Effect of Multiplying Coordinates by a Negative Number

When you multiply the coordinates of a figure by a negative number, you essentially perform a dilation with a negative scale factor. This has two primary effects:

  1. Scaling: The figure is scaled by the absolute value of the negative number. For example, multiplying coordinates by -2 would double the size of the figure.

  2. Reflection: The figure is reflected across the origin. This means that every point in the original figure is flipped across the origin, resulting in a mirror image.

Impact on Side Lengths, Perimeter, and Area

Let’s examine how multiplying coordinates by a negative number affects the side lengths, perimeter, and area of a geometric figure using a simple example: a rectangle.

Example: Rectangle

Consider a rectangle with vertices A(2, 1), B(4, 1), C(4, 3), and D(2, 3). Let’s multiply the coordinates of this rectangle by -2.

The new vertices after the transformation will be:

A'(-4, -2), B'(-8, -2), C'(-8, -6), and D'(-4, -6)

Side Lengths

  • Original Rectangle: AB = CD = 2 units, AD = BC = 2 units.
  • Transformed Rectangle: A’B’ = C’D’ = 4 units, A’D’ = B’C’ = 4 units.

Notice that the side lengths of the transformed rectangle are twice the original side lengths. This is because the scale factor of the dilation is 2 (the absolute value of -2).

Perimeter

  • Original Rectangle: Perimeter = 2(AB + AD) = 2(2 + 2) = 8 units.
  • Transformed Rectangle: Perimeter = 2(A’B’ + A’D’) = 2(4 + 4) = 16 units.

The perimeter of the transformed rectangle is also doubled, directly proportional to the change in side lengths.

Area

  • Original Rectangle: Area = AB * AD = 2 * 2 = 4 square units.
  • Transformed Rectangle: Area = A’B’ * A’D’ = 4 * 4 = 16 square units.

The area of the transformed rectangle is four times the original area. This is because the area is proportional to the square of the scale factor (2² = 4).

Generalization

The observations from the rectangle example can be generalized for any geometric figure:

  • Side Lengths: Multiplying coordinates by a negative number scales the side lengths by the absolute value of the negative number.
  • Perimeter: The perimeter of the transformed figure is scaled by the absolute value of the negative number.
  • Area: The area of the transformed figure is scaled by the square of the absolute value of the negative number.

Conclusion

Multiplying coordinates by a negative number results in a dilation with a negative scale factor. This transformation scales the figure by the absolute value of the negative number and reflects it across the origin. The side lengths, perimeter, and area of the figure are affected proportionally to the scale factor and its square, respectively. Understanding these concepts is essential for analyzing geometric transformations and their impact on the properties of geometric figures.

3. GeoGebra – Dilations

Citations

  1. 1. Khan Academy – Dilations
  2. 2. Math is Fun – Dilations

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