Unit 9 Transformations Homework 5 Dilations

Dilation is a transformation that alters the size of a figure without changing its shape, a fundamental concept in geometry that often presents unique challenges. Understanding how to perform and interpret dilations is crucial for mastering transformations in mathematics, and this guide will break down the intricacies of Unit 9 Transformations Homework 5, focusing specifically on dilations.

Understanding Dilations: The Basics

Dilation, at its core, involves resizing a geometric figure by a scale factor relative to a fixed point known as the center of dilation. This transformation can either enlarge or reduce the size of the original figure (pre-image) to create a new figure (image).

Key Components of Dilation

• Center of Dilation: The fixed point from which all points on the pre-image are scaled. The location of the center of dilation significantly affects the final position of the image.
• Scale Factor (k): A numerical value that determines the extent of enlargement or reduction.
  • If k > 1, the image is an enlargement of the pre-image.
  • If 0 < k < 1, the image is a reduction of the pre-image.
  • If k = 1, the image is congruent to the pre-image (no change in size).
  • If k is negative, the image is dilated and reflected across the center of dilation.
• Pre-image: The original figure before the dilation.
• Image: The new figure after the dilation.

Properties Preserved Under Dilation

Dilation preserves certain properties of the geometric figure, which are essential to recognize for solving problems accurately.

• Angle Measures: The angles in the image are congruent to the corresponding angles in the pre-image. Dilation does not alter the angles.
• Parallelism: If lines are parallel in the pre-image, they remain parallel in the image.
• Collinearity: If points are collinear in the pre-image, they remain collinear in the image.

However, it's crucial to note that dilation does not preserve distance or length. The lengths of the sides in the image are scaled by the scale factor k.

Performing Dilations: Step-by-Step Guide

To perform a dilation effectively, follow these steps:

1. Identify the Center of Dilation: Determine the coordinates of the center of dilation, often denoted as (a, b).

2. Determine the Scale Factor: Identify the scale factor k. This value will dictate whether the figure is enlarged or reduced.

3. Calculate the Distance: For each vertex of the pre-image, calculate the horizontal and vertical distance from the center of dilation. If a vertex has coordinates (x, y), the horizontal distance is |x - a|, and the vertical distance is |y - b|.

4. Apply the Scale Factor: Multiply both the horizontal and vertical distances by the scale factor k.

   • New horizontal distance = k * |x - a|
   • New vertical distance = k * |y - b|

5. Find the New Coordinates: Use the new distances to find the coordinates of the corresponding vertex in the image.

   • New x-coordinate = a + k * (x - a)
   • New y-coordinate = b + k * (y - b)

6. Repeat for All Vertices: Repeat steps 3-5 for all vertices of the pre-image to find the coordinates of all vertices in the image.

7. Draw the Image: Connect the new vertices to form the dilated image.

Example: Dilation with Center at the Origin

Consider a triangle with vertices A(1, 2), B(3, 4), and C(5, 1). We want to dilate this triangle by a scale factor of 2, with the center of dilation at the origin (0, 0).

1. Center of Dilation: (0, 0)

2. Scale Factor: k = 2

3. Calculate Distances and Apply Scale Factor:

   • A(1, 2):
     • Horizontal distance from (0, 0): |1 - 0| = 1
     • Vertical distance from (0, 0): |2 - 0| = 2
     • New horizontal distance: 2 * 1 = 2
     • New vertical distance: 2 * 2 = 4
     • New coordinates: A'(2, 4)
   • B(3, 4):
     • Horizontal distance from (0, 0): |3 - 0| = 3
     • Vertical distance from (0, 0): |4 - 0| = 4
     • New horizontal distance: 2 * 3 = 6
     • New vertical distance: 2 * 4 = 8
     • New coordinates: B'(6, 8)
   • C(5, 1):
     • Horizontal distance from (0, 0): |5 - 0| = 5
     • Vertical distance from (0, 0): |1 - 0| = 1
     • New horizontal distance: 2 * 5 = 10
     • New vertical distance: 2 * 1 = 2
     • New coordinates: C'(10, 2)

4. Draw the Image: Connect the vertices A'(2, 4), B'(6, 8), and C'(10, 2) to form the dilated triangle.

Example: Dilation with Center Not at the Origin

Consider a triangle with vertices P(2, 3), Q(4, 5), and R(6, 2). We want to dilate this triangle by a scale factor of 0.5, with the center of dilation at (1, 1).

1. Center of Dilation: (1, 1)

2. Scale Factor: k = 0.5

3. Calculate Distances and Apply Scale Factor:

   • P(2, 3):
     • Horizontal distance from (1, 1): |2 - 1| = 1
     • Vertical distance from (1, 1): |3 - 1| = 2
     • New horizontal distance: 0.5 * 1 = 0.5
     • New vertical distance: 0.5 * 2 = 1
     • New coordinates: P'(1 + 0.5, 1 + 1) = P'(1.5, 2)
   • Q(4, 5):
     • Horizontal distance from (1, 1): |4 - 1| = 3
     • Vertical distance from (1, 1): |5 - 1| = 4
     • New horizontal distance: 0.5 * 3 = 1.5
     • New vertical distance: 0.5 * 4 = 2
     • New coordinates: Q'(1 + 1.5, 1 + 2) = Q'(2.5, 3)
   • R(6, 2):
     • Horizontal distance from (1, 1): |6 - 1| = 5
     • Vertical distance from (1, 1): |2 - 1| = 1
     • New horizontal distance: 0.5 * 5 = 2.5
     • New vertical distance: 0.5 * 1 = 0.5
     • New coordinates: R'(1 + 2.5, 1 + 0.5) = R'(3.5, 1.5)

4. Draw the Image: Connect the vertices P'(1.5, 2), Q'(2.5, 3), and R'(3.5, 1.5) to form the dilated triangle.

Homework 5: Common Types of Dilation Problems

Unit 9 Transformations Homework 5 likely includes a variety of problems related to dilations. Here are some common types:

1. Finding the Image Coordinates: Given the coordinates of the pre-image, the center of dilation, and the scale factor, find the coordinates of the image.

2. Determining the Scale Factor: Given the coordinates of the pre-image and the image, find the scale factor of the dilation.

3. Finding the Center of Dilation: Given the coordinates of the pre-image, the image, and the scale factor, find the coordinates of the center of dilation.

4. Describing the Dilation: Given the pre-image and the image, describe the dilation, including the center of dilation and the scale factor.

5. Dilations in the Coordinate Plane: Problems involving plotting points and drawing figures on the coordinate plane to perform dilations.

6. Real-World Applications: Problems that apply the concept of dilation to real-world scenarios, such as map scaling or photography.

Strategies for Solving Dilation Problems

• Use the Formula: The general formula for finding the coordinates of the image after dilation is:

  • x' = a + k * (x - a)
  • y' = b + k * (y - b) where (x, y) are the coordinates of the pre-image, (a, b) are the coordinates of the center of dilation, k is the scale factor, and (x', y') are the coordinates of the image.

• Visualize the Transformation: Sketching the pre-image, the center of dilation, and a rough estimate of the image can help you understand the problem and avoid mistakes.

• Check Your Work: After performing the dilation, check that the image is scaled correctly and that the angles are preserved.

• Understand Proportionality: Dilation involves proportionality. The ratio of the distances between corresponding points in the pre-image and the image is equal to the scale factor.

Advanced Concepts in Dilation

To fully master dilations, it's important to understand some advanced concepts:

Negative Scale Factors

When the scale factor k is negative, the image is dilated and reflected across the center of dilation. This means that the image will be on the opposite side of the center of dilation from the pre-image.

• If k = -1, the transformation is a reflection through the center of dilation.
• If -1 < k < 0, the image is reduced and reflected.
• If k < -1, the image is enlarged and reflected.

Dilations and Similarity

Dilation is closely related to the concept of similarity. Two figures are similar if they have the same shape but different sizes. Dilation is a transformation that produces similar figures. If you dilate a pre-image, the resulting image is similar to the pre-image.

• Similar Figures: Figures that have the same shape but different sizes.
• Similarity Transformations: Transformations that produce similar figures, including dilations, reflections, rotations, and translations.

Composition of Transformations

Dilation can be combined with other transformations, such as translations, rotations, and reflections, to create more complex transformations. The order in which the transformations are applied can affect the final image.

• Translation: Shifts the figure without changing its size or shape.
• Rotation: Turns the figure around a fixed point.
• Reflection: Flips the figure across a line.

Common Mistakes to Avoid

• Incorrectly Applying the Scale Factor: Ensure you multiply the distances from the center of dilation by the scale factor, not the coordinates of the vertices.

• Forgetting the Center of Dilation: Always consider the center of dilation when calculating the new coordinates. If the center is not at the origin, you must account for its position.

• Misinterpreting Negative Scale Factors: Remember that a negative scale factor results in both dilation and reflection.

• Not Checking the Results: Verify that the image is correctly scaled and positioned relative to the pre-image and the center of dilation.

Real-World Applications of Dilation

Dilation is not just a theoretical concept; it has many practical applications in various fields:

1. Photography and Image Scaling: When you zoom in or out on a digital image, you are essentially performing a dilation.

2. Architecture and Engineering: Architects and engineers use dilations to create scaled models of buildings and structures.

3. Mapmaking: Maps are scaled-down versions of real-world geographic areas, which involves dilation.

4. Computer Graphics: Dilation is used in computer graphics to resize objects and create visual effects.

5. Fashion Design: Designers use dilations to create patterns and adjust the sizes of clothing.

Examples of Dilation Problems and Solutions

Problem 1: Finding Image Coordinates

A triangle has vertices A(2, 4), B(6, 2), and C(4, 6). Find the coordinates of the image after a dilation with a scale factor of 1.5 and the center of dilation at the origin (0, 0).

Solution:

Using the formula x' = a + k * (x - a) and y' = b + k * (y - b), with (a, b) = (0, 0) and k = 1.5:

• A(2, 4):
  • A'(1.5 * 2, 1.5 * 4) = A'(3, 6)
• B(6, 2):
  • B'(1.5 * 6, 1.5 * 2) = B'(9, 3)
• C(4, 6):
  • C'(1.5 * 4, 1.5 * 6) = C'(6, 9)

The coordinates of the image are A'(3, 6), B'(9, 3), and C'(6, 9).

Problem 2: Determining the Scale Factor

A square has vertices P(1, 1), Q(1, 3), R(3, 3), and S(3, 1). After a dilation with the center at the origin, the image has vertices P'(2, 2), Q'(2, 6), R'(6, 6), and S'(6, 2). Find the scale factor of the dilation.

Solution:

To find the scale factor, compare the coordinates of corresponding vertices. For example, compare P(1, 1) and P'(2, 2):

• x' = k * x => 2 = k * 1 => k = 2
• y' = k * y => 2 = k * 1 => k = 2

The scale factor of the dilation is 2.

Problem 3: Finding the Center of Dilation

Triangle ABC has vertices A(2, 3), B(4, 1), and C(6, 5). After a dilation with a scale factor of 0.5, the image has vertices A'(1.5, 2), B'(2.5, 1), and C'(3.5, 3). Find the center of dilation.

Solution:

Let the center of dilation be (a, b). We can set up equations using the dilation formula:

• A'(1.5, 2):
  • 1. 5 = a + 0.5 * (2 - a)
  • 2 = b + 0.5 * (3 - b)
• B'(2.5, 1):
  • 3. 5 = a + 0.5 * (4 - a)
  • 1 = b + 0.5 * (1 - b)

Solving equation 1:

1. 5 = a + 1 - 0.5a
2. 5 = 0.5a + 1
3. 5 = 0.5a a = 1

Solving equation 2: 2 = b + 1.5 - 0.5b 2 = 0.5b + 1.5 4. 5 = 0.5b b = 1

The center of dilation is (1, 1).

Conclusion

Mastering dilations is crucial for a strong foundation in geometry and transformations. By understanding the key components, following the step-by-step procedures, and practicing various types of problems, you can confidently tackle Unit 9 Transformations Homework 5 and any dilation-related challenges. Remember to visualize the transformations, double-check your calculations, and apply the concepts to real-world scenarios to deepen your understanding. Good luck!