unit 9 transformations homework 5 dilations answer key

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Unit 9 Transformations Homework 5: Dilation Answer Key – A Comprehensive Guide

This guide provides comprehensive answers and explanations for Unit 9, Homework 5, focusing on dilations. Understanding dilations is crucial for mastering geometric transformations. We'll break down the key concepts and provide solutions to help you solidify your understanding. Remember to always refer back to your textbook and class notes for additional support.

What is a Dilation?

A dilation is a transformation that changes the size of a figure, but not its shape. It's defined by a center point and a scale factor. The scale factor determines how much larger or smaller the figure becomes.

• Scale Factor > 1: The figure is enlarged.
• Scale Factor = 1: The figure remains unchanged.
• 0 < Scale Factor < 1: The figure is reduced.
• Scale Factor < 0: The figure is enlarged and reflected across the center point.

Key Concepts to Master Before Attempting Homework 5

Before diving into the answers, ensure you understand these critical concepts:

• Center of Dilation: The point from which the dilation occurs. All points are scaled relative to this center.
• Scale Factor (k): The ratio of the distance from the center to a point on the dilated image to the distance from the center to the corresponding point on the original image.
• Coordinate Rule for Dilations: If the center of dilation is the origin (0,0), the rule is simple: (x, y) → (kx, ky). If the center is not the origin, the process is slightly more complex, often involving translating the figure to the origin, performing the dilation, and then translating back.

Sample Problems and Solutions (Adapt to your specific homework problems)

Let's assume your homework includes problems like these. Remember to replace these with your actual problems.

Problem 1: Dilate triangle ABC with vertices A(2, 4), B(6, 2), and C(4, 0) by a scale factor of 2, using the origin as the center of dilation. Find the coordinates of the dilated triangle A'B'C'.

Solution 1:

Using the coordinate rule (x, y) → (2x, 2y), we get:

• A(2, 4) → A'(4, 8)
• B(6, 2) → B'(12, 4)
• C(4, 0) → C'(8, 0)

Therefore, the vertices of the dilated triangle A'B'C' are A'(4, 8), B'(12, 4), and C'(8, 0).

Problem 2: A square with vertices (1,1), (3,1), (3,3), and (1,3) is dilated using the origin as the center with a scale factor of 1/2. Find the coordinates of the image.

Solution 2:

Applying the coordinate rule (x,y) → (x/2, y/2):

• (1,1) → (1/2, 1/2)
• (3,1) → (3/2, 1/2)
• (3,3) → (3/2, 3/2)
• (1,3) → (1/2, 3/2)

Problem 3 (More Complex): Dilate the rectangle with vertices P(1,2), Q(4,2), R(4,5), S(1,5) using the point (2,1) as the center of dilation with a scale factor of 3.

Solution 3: This requires a more involved process:

1. Translate: Shift the rectangle so that the center of dilation is at the origin. Subtract (2,1) from each coordinate:

   • P'( -1, 1)
   • Q'(2, 1)
   • R'(2, 4)
   • S'(-1, 4)

2. Dilate: Apply the dilation with a scale factor of 3:

   • P''(-3, 3)
   • Q''(6, 3)
   • R''(6, 12)
   • S''(-3, 12)

3. Translate Back: Add (2,1) to each coordinate to return the rectangle to its original position:

   • P'''(-1, 4)
   • Q'''(8, 4)
   • R'''(8, 13)
   • S'''(-1, 13)

Tips for Success

• Draw diagrams: Visualizing the transformation makes it easier to understand.
• Check your work: Make sure your calculations are accurate.
• Practice: The more problems you solve, the better you'll understand dilations.

This guide provides a framework for approaching your dilation homework. Remember to adapt the examples and solutions to your specific problems. Good luck!