Lesson 4 Homework Practice Dilations Answer Key

Lesson 4 Homework Practice Dilations: A Comprehensive Guide with Answers

Understanding dilations is crucial in geometry, forming the foundation for further exploration of similar figures and transformations. This comprehensive guide delves into Lesson 4 Homework Practice on dilations, providing detailed explanations, step-by-step solutions, and insightful tips to master this concept. We'll cover various dilation scenarios, including those involving coordinates, scale factors, and different centers of dilation. Let's embark on this journey to conquer dilations!

Understanding Dilations: The Basics

Before we dive into the homework problems, let's refresh our understanding of dilations. A dilation is a transformation that enlarges or reduces a figure proportionally. It's defined by two key elements:

• Center of dilation: The fixed point around which the dilation occurs. Think of this as the "pivot point" of the transformation.

• Scale factor: A numerical value (k) that determines the extent of enlargement or reduction.

  • k > 1: The figure is enlarged (an enlargement).
  • 0 < k < 1: The figure is reduced (a reduction).
  • k = 1: The figure remains unchanged (the identity transformation).
  • k < 0: The figure is enlarged or reduced and reflected across the center of dilation.

Types of Dilations

Dilations can be categorized based on the location of the center of dilation:

• Dilations centered at the origin (0,0): These are often simpler to calculate because the coordinates of the dilated points are directly related to the original coordinates through multiplication by the scale factor.

• Dilations centered at a point other than the origin: These require a more systematic approach, often involving calculating vectors or using the distance formula.

Solving Dilation Problems: A Step-by-Step Approach

Let's tackle some common dilation problems encountered in Lesson 4 Homework Practice. Remember, the key is to understand the relationship between the original points and their dilated counterparts through the scale factor and center of dilation.

Problem Type 1: Finding the Coordinates of Dilated Points (Center at Origin)

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

Solution:

1. Multiply the x and y coordinates of each vertex by the scale factor:

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

2. The dilated triangle A'B'C' has vertices A'(4, 8), B'(12, 4), and C'(8, 0).

Problem Type 2: Finding the Coordinates of Dilated Points (Center Not at Origin)

Problem: A rectangle with vertices P(1,1), Q(3,1), R(3,3), and S(1,3) is dilated with a scale factor of 1.5 and center of dilation at (0,2). Find the coordinates of the dilated rectangle P'Q'R'S'.

Solution: This requires a more involved process. We'll use the concept of vectors.

1. Find the vector from the center of dilation to each vertex:

   • Vector from (0,2) to P(1,1): <1-0, 1-2> = <1, -1>
   • Vector from (0,2) to Q(3,1): <3-0, 1-2> = <3, -1>
   • Vector from (0,2) to R(3,3): <3-0, 3-2> = <3, 1>
   • Vector from (0,2) to S(1,3): <1-0, 3-2> = <1, 1>

2. Multiply each vector by the scale factor (1.5):

   • 1.5*<1,-1> = <1.5, -1.5>
   • 1.5*<3,-1> = <4.5, -1.5>
   • 1.5*<3,1> = <4.5, 1.5>
   • 1.5*<1,1> = <1.5, 1.5>

3. Add these scaled vectors to the center of dilation (0,2) to find the coordinates of the dilated vertices:

   • P'(0+1.5, 2-1.5) = P'(1.5, 0.5)
   • Q'(0+4.5, 2-1.5) = Q'(4.5, 0.5)
   • R'(0+4.5, 2+1.5) = R'(4.5, 3.5)
   • S'(0+1.5, 2+1.5) = S'(1.5, 3.5)

4. The dilated rectangle P'Q'R'S' has vertices P'(1.5, 0.5), Q'(4.5, 0.5), R'(4.5, 3.5), and S'(1.5, 3.5).

Problem Type 3: Finding the Scale Factor

Problem: Triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 2) is dilated to triangle A'B'C' with vertices A'(2, 4), B'(6, 8), and C'(10, 4). Find the scale factor.

Solution:

1. Choose corresponding vertices from the original and dilated triangles: Let's use A(1,2) and A'(2,4).

2. Compare the x-coordinates: The x-coordinate of A' is twice the x-coordinate of A (2/1 = 2).

3. Compare the y-coordinates: The y-coordinate of A' is twice the y-coordinate of A (4/2 = 2).

4. Since both ratios are the same, the scale factor is 2. You can verify this by comparing the other vertices.

Problem Type 4: Determining if Figures are Dilations of Each Other

Problem: Determine if quadrilateral ABCD with vertices A(1,1), B(3,1), C(3,3), and D(1,3) is a dilation of quadrilateral A'B'C'D' with vertices A'(2,2), B'(6,2), C'(6,6), and D'(2,6). If so, find the scale factor and center of dilation.

Solution:

1. Compare corresponding sides: Notice that each side of A'B'C'D' is twice the length of the corresponding side of ABCD. This suggests a dilation.

2. Check the ratio of corresponding coordinates: The coordinates of A'B'C'D' are double the coordinates of ABCD. This strongly suggests a dilation with a scale factor of 2.

3. Determine the center of dilation: Since the dilation is by a factor of 2 from the origin, the center of dilation is likely the origin.

Advanced Dilation Problems and Considerations

Lesson 4 might also include more challenging problems involving:

• Dilations in three dimensions: These problems extend the concept to three-dimensional figures, requiring work with three coordinates (x, y, z).

• Compositions of dilations: This involves performing multiple dilations sequentially, requiring careful tracking of the transformations.

• Dilations and other transformations: Problems might combine dilations with rotations, reflections, or translations.

Tips for Mastering Dilations

• Visualize: Draw diagrams to represent the original and dilated figures. This will help you understand the transformation process.

• Organize your work: Keep track of your calculations neatly. This will help you avoid errors and make it easier to check your answers.

• Practice regularly: The best way to master dilations is through consistent practice. Work through as many problems as you can to build your understanding and skills.

• Seek help when needed: Don't hesitate to ask your teacher, tutor, or classmates for help if you get stuck.

This comprehensive guide covers various aspects of Lesson 4 Homework Practice on dilations. By understanding the fundamentals, applying systematic approaches to problem-solving, and consistently practicing, you'll confidently navigate these geometric transformations and achieve mastery. Remember, practice is key, and with focused effort, you can conquer the challenges presented in your homework. Good luck!