Understanding Dilation: A Step‑by‑Step Guide to Unit 9 Homework 5
Dilation is one of the most powerful tools in the geometry toolkit. In Unit 9, Homework 5 focuses on applying dilation rules, finding scale factors, and determining the images of key points. It lets you stretch or shrink a figure while keeping its shape intact, simply by scaling every point relative to a fixed center. This article walks through the concepts, offers strategies for tackling each problem, and provides a clear, practice‑ready framework that will help you ace the assignment.
Introduction
When you dilate a figure, you multiply every coordinate by the same constant—called the scale factor—with respect to a chosen center. If the scale factor is greater than 1, the figure grows; if it’s between 0 and 1, the figure shrinks; and if it’s negative, the figure also reflects across the center. Homework 5 asks you to identify these scale factors, compute new coordinates, and verify that the resulting shape is similar to the original.
1. Key Concepts Recap
| Concept | Definition | Formula |
|---|---|---|
| Dilation | A transformation that enlarges or reduces a figure by a scale factor. | |
| Scale factor ((k)) | The ratio of a side length in the image to the corresponding side length in the pre‑image. Here's the thing — | ( (x', y') = (k(x - h) + h,; k(y - k) + k) ) where ((h,k)) is the center and (k) is the scale factor. |
| Similarity | Two figures are similar if all corresponding angles are equal and all corresponding side lengths are in proportion. |
2. Step‑by‑Step Strategy for Each Problem
Step 1: Identify the Center of Dilation
- Look for a point labeled (O), (C), or a specific coordinate that is explicitly mentioned in the problem.
- If the center isn’t given, check the diagram for a point that all lines in the problem seem to radiate from.
Step 2: Determine the Scale Factor
- Case A: Side lengths are given
- Pick any pair of corresponding sides (pre‑image and image).
- Compute (k = \frac{\text{image side}}{\text{pre‑image side}}).
- Case B: Coordinates are given
- Use the distance formula to find the distance from the center to a point in the pre‑image and the distance from the center to the corresponding image point.
- Then (k = \frac{\text{distance to image point}}{\text{distance to pre‑image point}}).
Step 3: Apply the Dilation Formula
- For each point ((x, y)) in the pre‑image, compute its image ((x', y')) using: [ x' = h + k(x-h), \quad y' = k + k(y-k) ] where ((h,k)) is the center.
Step 4: Verify Similarity
- Check that all angles are preserved (usually obvious if coordinates are correct).
- Verify that the ratio of any two side lengths in the image equals the ratio in the pre‑image.
Step 5: Answer the Question
- Many problems ask for the coordinates of a specific point or the length of a side after dilation.
- Use the formulas above to compute the exact values.
3. Example Problems from Homework 5
Problem 1: Dilation Center at the Origin
Given: Triangle (ABC) with vertices (A(2, 3)), (B(5, 7)), (C(1, 1)). Dilation centered at the origin with a scale factor of 3.
Find: Coordinates of the image triangle (A'B'C') Simple, but easy to overlook..
Solution:
- Center (O(0,0)).
- Scale factor (k = 3).
- Apply formula:
- (A' = (3 \cdot 2, 3 \cdot 3) = (6, 9))
- (B' = (3 \cdot 5, 3 \cdot 7) = (15, 21))
- (C' = (3 \cdot 1, 3 \cdot 1) = (3, 3))
- Verify: side lengths multiplied by 3, angles unchanged.
Answer: (A'(6,9)), (B'(15,21)), (C'(3,3)).
Problem 2: Dilation Center Not at the Origin
Given: Quadrilateral (PQRS) with coordinates (P(1,2)), (Q(4,2)), (R(4,5)), (S(1,5)). Dilation centered at (C(2,3)) with scale factor (k = \frac{1}{2}).
Find: Coordinates of the image quadrilateral (P'Q'R'S') It's one of those things that adds up..
Solution:
- Center (C(2,3)).
- For each point, compute:
- (P': x' = 2 + \frac{1}{2}(1-2) = 2 - 0.5 = 1.5)
(y' = 3 + \frac{1}{2}(2-3) = 3 - 0.5 = 2.5) - Repeat for (Q, R, S).
- (P': x' = 2 + \frac{1}{2}(1-2) = 2 - 0.5 = 1.5)
- Final coordinates:
- (P'(1.5, 2.5))
- (Q'(3.5, 2.5))
- (R'(3.5, 4.5))
- (S'(1.5, 4.5))
Answer: (P'(1.5, 2.5)), (Q'(3.5, 2.5)), (R'(3.5, 4.5)), (S'(1.5, 4.5)) Surprisingly effective..
Problem 3: Finding the Scale Factor from Side Lengths
Given: Triangle (TUV) before dilation: (TU = 4), (UV = 5), (VT = 3). After dilation, side (T'U' = 12).
Find: Scale factor (k) and the lengths of (U'V') and (V'T').
Solution:
- Since (T'U' = 12) corresponds to (TU = 4),
(k = \frac{12}{4} = 3). - Apply (k) to other sides:
- (U'V' = 5k = 15)
- (V'T' = 3k = 9)
Answer: (k = 3); (U'V' = 15); (V'T' = 9) No workaround needed..
4. Common Mistakes and How to Avoid Them
| Mistake | Why It Happens | Fix |
|---|---|---|
| Using the wrong center | Center not explicitly stated | Double‑check the diagram or problem statement |
| Mixing up pre‑image and image points | Labels swapped | Label points clearly before calculation |
| Forgetting to apply the sign of (k) | Negative scale factor ignored | Include the negative sign in the formula |
| Rounding prematurely | Early rounding leads to errors | Keep fractions or decimals until the final answer |
5. Frequently Asked Questions
Q1: What if the dilation center is at a point that isn’t the origin?
A: Use the general dilation formula: (x' = h + k(x-h)), (y' = k + k(y-k)), where ((h,k)) is the center.
Q2: How do I verify that the image is indeed similar to the original?
A: Check that all corresponding side ratios equal the scale factor and that the angles remain unchanged. A quick visual inspection of the diagram can also confirm similarity Worth keeping that in mind..
Q3: Can the scale factor be negative?
A: Yes. A negative scale factor reflects the figure across the center while scaling it. The absolute value of (k) indicates the size change, while the sign indicates reflection Most people skip this — try not to..
Q4: What if only one side length is given after dilation?
A: Use that side to compute the scale factor and apply it to the other sides or coordinates It's one of those things that adds up. Practical, not theoretical..
Q5: How do I handle fractional coordinates after dilation?
A: Keep them as fractions or decimals until the final answer. If the problem asks for integer coordinates, double‑check the calculations.
6. Practice Tips for Mastery
- Draw the figures: Sketching pre‑image and image helps visualize the scaling and center.
- Work with vectors: Treat each point as a vector from the center; dilation is simply scaling that vector.
- Check units: If side lengths are in centimeters, keep units consistent throughout.
- Cross‑verify: Compute the scale factor using two different side pairs; they should match.
- Use a calculator: For non‑integer coordinates, a scientific calculator or spreadsheet ensures accuracy.
Conclusion
Dilation transforms geometry by scaling figures while preserving shape and angles. By mastering the identification of the center, calculation of the scale factor, and application of the dilation formula, you can confidently solve any problem in Unit 9, Homework 5. In practice, remember to verify similarity, watch for common pitfalls, and practice with diverse examples. With these skills sharpened, you’ll not only ace the homework but also build a solid foundation for more advanced geometric transformations And it works..
