Dilations Not Centered At The Origin Worksheet

Introduction

This dilations not centered at the origin worksheet provides a step‑by‑step guide for students learning how to perform dilations when the center of dilation is any point other than the origin. You will discover clear instructions, visual examples, and practice problems that reinforce the concepts of scale factor, center of dilation, and coordinate transformation. By the end of the worksheet, learners will be able to plot dilated figures, calculate missing lengths, and explain the geometric reasoning behind non‑centered dilations with confidence Worth keeping that in mind..

Understanding Dilations Not Centered at the Origin

What is a Dilation?

A dilation is a transformation that produces a figure similar to the original by expanding or shrinking it proportionally. The key elements are the center of dilation (the fixed point from which the figure is enlarged or reduced) and the scale factor (the ratio of the new length to the original length).

Difference from Centered Dilations

When the center is the origin (0, 0), the mathematics simplify because every point (x, y) moves along a line through the origin. Dilations not centered at the origin require an additional step: translate the figure so that the chosen center becomes the origin, apply the dilation, then translate back. This extra translation makes the process slightly more complex but follows the same proportional rules.

Steps to Perform a Dilation Not Centered at the Origin

1. Identify the center of dilation (C) and the scale factor (k).
2. Translate each vertex of the original figure so that C moves to the origin:
   • For a point P(x, y), compute the translated coordinates:
     [ P' = (x - C_x,; y - C_y) ]
3. Apply the dilation using the scale factor k:
   • New coordinates after dilation:
     [ P'' = (k \cdot P'_x,; k \cdot P'_y) ]
4. Translate back to the original coordinate system by adding the center coordinates:
   • Final point:
     [ P_{\text{new}} = (P''_x + C_x,; P''_y + C_y) ]
5. Connect the new points to obtain the dilated figure.

Tip: Keep a table of original points, translated points, dilated points, and final points to avoid errors.

Worked Example Problems

Example 1 – Triangle Dilation

Problem: Dilate triangle ABC with vertices A(1, 2), B(4, 5), C(2, 7) by a scale factor of 2, centered at point C(2, 7) Nothing fancy..

Solution:

• Step 1: Center C is (2, 7).
• Step 2: Translate each point so C becomes the origin:
  • A': (1‑2, 2‑7) = (‑1, ‑5)
  • B': (4‑2, 5‑7) = (2, ‑2)
  • C' = (0, 0) (since it is the center)
• Step 3: Multiply by k = 2:
  • A'' = (‑1 × 2, ‑5 × 2) = (‑2, ‑10)
  • B'' = (2 × 2, ‑2 × 2) = (4, ‑4)
  • C'' = (0, 0)
• Step 4: Translate back by adding C(2, 7):
  • A_new = (‑2 + 2, ‑10 + 7) = (0, ‑3)
  • B_new = (4 + 2, ‑4 + 7) = (6, 3)
  • C_new = (0 + 2, 0 + 7) = (2, 7)

The dilated triangle has vertices (0, ‑3), (6, 3), and (2, 7).

Example 2 – Quadrilateral with Fractional Scale Factor

Problem: Dilate quadrilateral PQRST with points P(0, 0), Q(3, 1), R(5, 4), S(2, 6), T(‑1, 2) by a scale factor of ½, centered at point R(5, 4).

Solution:

• Step 1: Center R = (5, 4).
• Step 2: Translate points:
  • P' = (0‑5, 0‑4) = (‑5, ‑4)
  • Q' = (3‑5, 1‑4) = (‑2, ‑3)
  • R' = (0, 0)
  • S' = (2‑5, 6‑4) = (‑3, 2)
  • T' = (‑1‑5, 2‑4) = (‑6, ‑2)
• Step 3: Multiply by k = ½:
  • P'' = (‑5 × ½, ‑4