How to Perform Similarity Transformations: Step-by-Step Guide

Step-by-step Guide: Similarity Transformations

Basics of Transformations:
Transformations in geometry refer to operations that alter the position, size, or orientation of a figure. These can include translations (slides), rotations (turns), reflections (flips), and dilations (resizing).

Defining Similarity Transformations:
Similarity transformations are a subset of transformations that change the size of a figure without changing its shape. The primary similarity transformation is dilation.

Dilation – The Core of Similarity Transformation:
Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. It involves enlarging or reducing a figure proportionally.

• Center of Dilation: A fixed point in the plane.
• Scale Factor: The ratio by which a figure is enlarged or reduced. A scale factor greater than \(1\) results in an enlargement, while a scale factor between \(0\) and \(1\) results in a reduction. The formula for dilation from a center point \((x, y)\) with a scale factor \(k\) is:
  \( (x’, y’) = (x \cdot k, y \cdot k) \)

Examples

Example 1:
A rectangle has vertices at \(A(2,3)\), \(B(8,3)\), \(C(8,6)\), and \(D(2,6)\). Find the vertices of the rectangle after a dilation with a scale factor of \(2\), centered at the origin.

Solution:
Using the dilation formula:
\( A'(4,6), B'(16,6), C'(16,12), ) and ( D'(4,12) \).

Example 2:
A triangle has vertices at \(P(1,1)\), \(Q(4,1)\), and \(R(2,5)\). Determine the vertices of the triangle after a dilation with a scale factor of \(0.5\), centered at the origin.

Solution:
Applying the dilation formula:
\( P'(0.5,0.5), Q'(2,0.5), \) and \( R'(1,2.5) \).

Practice Questions:

1. A square has vertices at \(E(3,4)\), \(F(6,4)\), \(G(6,7)\), and \(H(3,7)\). What are the vertices of the square after a dilation with a scale factor of \(3\), centered at the origin?
2. A rhombus has vertices at \(I(-2,3)\), \(J(2,5)\), \(K(4,3)\), and \(L(0,1)\). Determine the vertices of the rhombus after a dilation with a scale factor of \(0.25\), centered at the origin.

Answers:

1. \(E'(9,12)\), \(F'(18,12)\), \(G'(18,21)\), and \(H'(9,21)\)
2. \(I'(-0.5,0.75)\), \(J'(0.5,1.25)\), \(K'(1,0.75)\), and \(L'(0,0.25)\)