This document provides potential answers for a hypothetical Unit 9 Transformations Homework 5 focusing on dilations. Since I do not have access to your specific assignment, these answers serve as a guide and should be compared to the instructions and diagrams provided in your actual homework. Always refer to your teacher's key or instructions for the definitive answers. Accuracy depends heavily on the specific details of your problems.
Understanding Dilations
Before diving into the answers, let's quickly review the key concepts of dilations:
- Center of Dilation: The point from which the dilation occurs. All points are scaled relative to this center.
- Scale Factor: The ratio of the distance from the center of dilation to a point on the dilated image to the distance from the center of dilation to the corresponding point on the original image. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction.
Sample Problems and Potential Answers
The following are examples of problems commonly found in a unit on dilations, along with potential solutions. Remember to adapt these examples to your specific problems.
Problem 1: A triangle with vertices A(1,1), B(3,1), C(2,3) is dilated with a center of dilation at the origin (0,0) and a scale factor of 2. Find the coordinates of the vertices of the dilated triangle A'B'C'.
Answer 1: To find the coordinates of the dilated triangle, multiply the coordinates of each vertex by the scale factor:
- A'(2,2) (12, 12)
- B'(6,2) (32, 12)
- C'(4,6) (22, 32)
Problem 2: A square with side length 4 units is dilated by a scale factor of 0.5. What is the side length of the dilated square?
Answer 2: Multiply the original side length by the scale factor: 4 units * 0.5 = 2 units. The side length of the dilated square is 2 units.
Problem 3: Describe the transformation that maps triangle ABC to triangle A'B'C', given the coordinates: A(1,2), B(3,4), C(2,5) and A'(3,6), B'(9,12), C'(6,15).
Answer 3: Notice that the coordinates of A', B', and C' are three times the coordinates of A, B, and C respectively. This indicates a dilation with a scale factor of 3. To determine the center, we would need further information or a graph to see if the dilation is centered at the origin or another point.
Problem 4: A circle with a radius of 5 cm is dilated with a scale factor of 3. What is the radius of the dilated circle?
Answer 4: The radius of the dilated circle is 5 cm * 3 = 15 cm.
Problem 5: (This would involve a graphical problem). You are given a graph showing a pre-image and a dilated image. You need to identify the center of dilation and the scale factor.
Answer 5: (This answer would depend entirely on the graphical information provided. You would measure distances from the center of dilation to corresponding points on the pre-image and the image to calculate the scale factor.)
Important Considerations:
- Graphing: Many dilation problems are best solved using graph paper. Plotting the points and visualizing the transformation can greatly aid understanding.
- Negative Scale Factors: Remember that a negative scale factor implies a dilation combined with a reflection across the center of dilation.
- Different Centers of Dilation: Problems might use a center of dilation other than the origin. Make sure to account for this when performing calculations.
Remember to replace these sample problems and answers with the actual problems from your homework. If you have specific questions about particular problems, provide the details, and I will do my best to help. Good luck!
