Unit 9 Transformations Homework 5 Dilations Answer Key

Unit 9 Transformations Homework 5 Dilations: A Comprehensive Guide

Finding the answer key for your homework can be tricky, especially when dealing with complex mathematical concepts like dilations. This comprehensive guide will delve into the core principles of dilations, providing you with a solid understanding to tackle Unit 9 Transformations Homework 5 and similar problems independently. We'll move beyond simply providing answers and focus on building your problem-solving skills. Remember, true understanding surpasses simply getting the right answer.

Understanding Dilations: The Foundation

Before we jump into solving problems, let's establish a firm grasp on the concept of dilation. A dilation is a transformation that changes the size of a figure, but not its shape. It's like enlarging or reducing a photo – the image remains the same, just scaled up or down. This scaling factor is called the scale factor, often denoted by 'k'.

Key Components of a Dilation:

• Scale Factor (k): This determines the size change.

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

• Center of Dilation: This is the fixed point around which the dilation occurs. All points are scaled relative to this center. It can be a point inside the figure, outside the figure, or even on the figure itself.

How Dilations Affect Coordinates:

The coordinates of the dilated figure are found by multiplying the original coordinates by the scale factor. Let's say we have a point (x, y) and a scale factor k. The new coordinates (x', y') after dilation will be:

(x', y') = (kx, ky)

This is the crucial formula for solving dilation problems. Remember that the center of dilation remains fixed.

Tackling Unit 9 Transformations Homework 5: A Step-by-Step Approach

Let's assume your homework involves various problems related to dilations. We'll break down common problem types and demonstrate a methodical approach.

Problem Type 1: Finding the Coordinates of a Dilated Figure

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

Solution:

1. Identify the Scale Factor (k) and Center of Dilation: Here, k = 2, and the center is (0, 0).

2. Apply the Dilation Formula: For each vertex, multiply the x and y coordinates by the scale factor:

   • A'(22, 42) = A'(4, 8)
   • B'(62, 22) = B'(12, 4)
   • C'(42, 02) = C'(8, 0)

3. State the Answer: The coordinates of the dilated triangle are A'(4, 8), B'(12, 4), and C'(8, 0).

Problem Type 2: Determining the Scale Factor

Problem Example: A rectangle with vertices P(1, 1), Q(3, 1), R(3, 2), and S(1, 2) is dilated to form a new rectangle P'Q'R'S' with vertices P'(2, 2), Q'(6, 2), R'(6, 4), and S'(2, 4). Find the scale factor of the dilation.

Solution:

1. Choose a Corresponding Pair of Vertices: Let's use P(1, 1) and P'(2, 2).

2. Find the Ratio: Divide the x-coordinate of the dilated vertex by the x-coordinate of the original vertex (and do the same for the y-coordinates). This ratio should be the same for both x and y.

   • x-ratio: 2/1 = 2
   • y-ratio: 2/1 = 2

3. State the Answer: The scale factor (k) is 2.

Problem Type 3: Dilations with a Center of Dilation Other Than the Origin

Problem Example: A square with vertices D(1, 1), E(3, 1), F(3, 3), and G(1, 3) is dilated with a scale factor of k = 1/2 and a center of dilation at point C(0, 2). Find the coordinates of the dilated square D'E'F'G'. This problem requires a slightly more complex approach.

Solution: This problem requires a vector approach.

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

   • Vector CD = D - C = (1-0, 1-2) = (1, -1)
   • Vector CE = E - C = (3-0, 1-2) = (3, -1)
   • Vector CF = F - C = (3-0, 3-2) = (3, 1)
   • Vector CG = G - C = (1-0, 3-2) = (1, 1)

2. Multiply each vector by the scale factor:

   • Vector CD' = (1/2)(1, -1) = (1/2, -1/2)
   • Vector CE' = (1/2)(3, -1) = (3/2, -1/2)
   • Vector CF' = (1/2)(3, 1) = (3/2, 1/2)
   • Vector CG' = (1/2)(1, 1) = (1/2, 1/2)

3. Add the center of dilation back to each resulting vector:

   • D' = C + CD' = (0 + 1/2, 2 - 1/2) = (1/2, 3/2)
   • E' = C + CE' = (0 + 3/2, 2 - 1/2) = (3/2, 3/2)
   • F' = C + CF' = (0 + 3/2, 2 + 1/2) = (3/2, 5/2)
   • G' = C + CG' = (0 + 1/2, 2 + 1/2) = (1/2, 5/2)

4. State the answer: The coordinates of the dilated square are D'(1/2, 3/2), E'(3/2, 3/2), F'(3/2, 5/2), and G'(1/2, 5/2).

Problem Type 4: Word Problems Involving Dilations

Word problems test your understanding of the concepts in a real-world context. Always break down the problem into manageable steps. Identify the key information (scale factor, center of dilation, original figure), apply the dilation formula, and solve for the unknown.

Beyond the Answer Key: Mastering Dilations

Obtaining the answer key might provide immediate gratification, but genuine understanding comes from actively engaging with the problems. Here's how to go beyond simply finding the answers:

• Practice Regularly: The more problems you solve, the more comfortable you'll become with applying the concepts.

• Visualize the Transformations: Sketch the original and dilated figures. This helps you understand the effect of the scale factor and center of dilation.

• Understand the Underlying Principles: Don't just memorize formulas; try to grasp the geometric meaning behind dilations.

• Seek Help When Needed: Don't hesitate to ask your teacher, classmates, or online resources for clarification if you encounter difficulties.

• Check Your Work: Always verify your answers using different methods or by comparing them to your sketches.

This detailed guide provides a comprehensive approach to solving dilation problems from Unit 9 Transformations Homework 5. By understanding the core principles and applying the steps outlined, you'll not only find the correct answers but develop a strong foundational knowledge of dilations that will serve you well in your future mathematical endeavors. Remember, the focus should be on learning and understanding, not just getting the answers. Good luck!