Dilation Transformation of Quadrilateral QRTW

Graph and Dilation Concepts

Introduction to Dilation

• Definition: A dilation is a transformation that alters the size of a figure but retains its shape. This transformation occurs relative to a fixed point known as the center of dilation.
• Center of Dilation: In the scenario presented, the center of dilation is at the origin (0, 0).
• Scale Factor: The scale factor describes how much the figure will be enlarged or reduced. In this case, the scale factor is given as 13.

Understanding the Scale Factor

• A scale factor greater than 1 results in enlargement of the figure.
• A scale factor less than 1 (but greater than 0) results in a reduction of size.
• The transformation of any point can be expressed by the formula:
  • If the original point is given by an ordered pair ((x, y)), after dilation, the new point ((x', y')) will be calculated using the formula:
    $\begin{cases} x' = kx \ y' = ky \end{cases}$
    Where (k) is the scale factor.

Dilation Calculation for Vertices of QRTW

Given Vertices of Quadrilateral QRTW

• For the dilation transformation, the vertices of the quadrilateral QRTW at their original positions must be known; assume these vertices are:
  • Q(x₁, y₁)
  • R(x₂, y₂)
  • T(x₃, y₃)
  • W(x₄, y₄)

Application of the Dilation Formula

1. Finding New Coordinates: For each vertex, apply the dilation formula with a scale factor of 13.
   • Vertex Q:
     $\begin{cases} Q' = (13x<em>1, 13y</em>1) \end{cases}$
   • Vertex R:
     $\begin{cases} R' = (13x<em>2, 13y</em>2) \end{cases}$
   • Vertex T:
     $\begin{cases} T' = (13x<em>3, 13y</em>3) \end{cases}$
   • Vertex W:
     $\begin{cases} W' = (13x<em>4, 13y</em>4) \end{cases}$

Simplifying Coordinates

• After calculating each new coordinate for Q', R', T', and W', ensure to simplify the ordered pairs accordingly based on the values of the original vertices provided in the graph.

Final Outcome

• The final output for each vertex after dilation will yield vertices:
  • New Vertex Q':
  • New Vertex R':
  • New Vertex T':
  • New Vertex W':
• Conclusion: The new vertices form the image of QRTW after dilation about the origin with a scale factor of 13.
• Note: Specific ordered pairs will be noted after applying values from the vertices Q, R, T, and W accordingly once more details are provided from the graph to input specific coordinate values.