Dilations

Definition

Dilations are transformations that produce an image that is the same shape as the original figure but is a different size. This transformation involves enlarging or reducing a figure by a scale factor relative to a fixed point called the center of dilation, and it preserves the angles of the original figure while changing the lengths of the sides proportionally.

5 Must Know Facts For Your Next Test

1. Dilations can be defined using a center of dilation and a scale factor, where a scale factor greater than 1 results in an enlargement and a scale factor between 0 and 1 results in a reduction.
2. The distances from the center of dilation to corresponding points on the original figure and the dilated figure maintain a constant ratio equal to the scale factor.
3. Angles within the figure remain unchanged during a dilation, meaning all corresponding angles between the original and dilated figures are congruent.
4. When performing dilations on the coordinate plane, each point's coordinates are multiplied by the scale factor relative to the center of dilation.
5. Dilations play a key role in proving similarity between triangles, as they establish that corresponding sides are proportional and angles are equal.

Review Questions

• How do you determine whether two figures are similar using dilations?
  • To determine if two figures are similar using dilations, you must check if there is a scale factor that relates the lengths of corresponding sides. If all pairs of corresponding sides have lengths that maintain a consistent ratio, then the figures are similar. Additionally, you must verify that all corresponding angles are equal, which confirms that dilations maintain angle measures.
• What role do scale factors play in dilations, and how do they affect the dimensions of a figure?
  • Scale factors are crucial in dilations as they dictate how much a figure will be enlarged or reduced. A scale factor greater than 1 enlarges the figure while keeping its shape intact, whereas a scale factor between 0 and 1 reduces its size. The application of the scale factor affects all dimensions proportionally, ensuring that both length and width change at the same rate, maintaining similarity between the original and dilated figures.
• Analyze how dilations can be applied to real-world problems involving similar figures and their properties.
  • Dilations can be applied in various real-world scenarios such as architectural design or map scaling, where it’s essential to maintain proportionality. For example, when creating blueprints for buildings, architects often use dilations to ensure that every feature remains proportional to its intended size. This principle allows for accurate representation of structures while enabling easy calculation of dimensions based on scaled models. Understanding how dilations work helps in making precise adjustments while preserving overall design integrity.

Related terms

Scale Factor: A number that scales, or multiplies, the dimensions of a figure during a dilation, determining how much larger or smaller the image will be compared to the original.

Similar Figures: Figures that have the same shape but may have different sizes, with corresponding angles equal and corresponding sides in proportion.

Transformation: A function that maps a figure onto another figure, which includes translations, rotations, reflections, and dilations.