Dilations/Translations Worksheet Answer Key - Quick Guide
Table of Contents :
Dilations and translations are fundamental concepts in geometry, primarily focusing on how shapes can be manipulated in a coordinate plane. Understanding these transformations is essential for students, especially when working on exercises or worksheets designed to test their knowledge. This article serves as a quick guide to help you navigate the concept of dilations and translations, along with an answer key for typical problems you might encounter.
What are Dilations?
Dilations are transformations that alter the size of a shape without changing its proportions. This means that a shape can be enlarged or reduced while retaining its original shape characteristics.
Key Concepts of Dilations
- Scale Factor: This is the ratio that determines how much the shape will be enlarged or reduced. A scale factor greater than 1 results in an enlargement, while a scale factor less than 1 leads to a reduction.
- Center of Dilation: This is a fixed point in the plane about which the dilation occurs. Every point in the shape is moved away from or towards this center depending on the scale factor.
Example of a Dilation
Consider a triangle with vertices at A(1, 2), B(3, 4), and C(5, 1) being dilated from the center O(0, 0) with a scale factor of 2.
- New vertex A’ = (1 * 2, 2 * 2) = (2, 4)
- New vertex B’ = (3 * 2, 4 * 2) = (6, 8)
- New vertex C’ = (5 * 2, 1 * 2) = (10, 2)
This dilation transforms the triangle into a larger triangle A’B’C’.
What are Translations?
Translations, on the other hand, involve sliding a shape from one position to another without changing its size, shape, or orientation.
Key Concepts of Translations
- Direction: Translations can be described in terms of how far and in which direction a shape moves on the coordinate plane.
- Vector Notation: The translation can be represented by a vector (a, b) where 'a' indicates the horizontal shift and 'b' indicates the vertical shift.
Example of a Translation
If you have a rectangle with vertices D(2, 3), E(2, 5), F(4, 5), and G(4, 3) translated by the vector (3, 2):
- New vertex D’ = (2 + 3, 3 + 2) = (5, 5)
- New vertex E’ = (2 + 3, 5 + 2) = (5, 7)
- New vertex F’ = (4 + 3, 5 + 2) = (7, 7)
- New vertex G’ = (4 + 3, 3 + 2) = (7, 5)
The rectangle is simply shifted to a new location.
Key Characteristics of Transformations
| Transformation Type | Definition | Size Change | Orientation Change |
|---|---|---|---|
| Dilation | Resizing around a center | Yes | No |
| Translation | Sliding to a new position | No | No |
Sample Worksheet Problems and Answer Key
Below are some example problems one might find on a worksheet regarding dilations and translations, along with their answers.
Problems
-
Dilation Problem: Find the coordinates of the image of point P(2, 3) under a dilation centered at the origin with a scale factor of 3.
-
Translation Problem: Translate the point Q(5, -2) by the vector (4, 6). What are the new coordinates?
-
Dilation Problem: A square has vertices A(1, 1), B(1, 2), C(2, 2), and D(2, 1). Find the vertices after a dilation with center O(0, 0) and a scale factor of 0.5.
-
Translation Problem: A triangle has vertices X(3, 1), Y(4, 5), and Z(5, 3). Translate this triangle by the vector (-2, 3).
Answers
-
Dilation:
New coordinates of P' = (2 * 3, 3 * 3) = (6, 9) -
Translation:
New coordinates of Q' = (5 + 4, -2 + 6) = (9, 4) -
Dilation:
New vertices:
A'(0.5, 0.5), B'(0.5, 1), C'(1, 1), D'(1, 0.5) -
Translation:
New coordinates:
X'(1, 4), Y'(2, 8), Z'(3, 6)
Important Notes
- Remember that in dilations, all points move towards or away from the center by the scale factor.
- For translations, every point of the shape moves the same distance and in the same direction as specified by the vector.
- It’s beneficial to practice both dilations and translations through various exercises to solidify your understanding.
By mastering these concepts, students will find it easier to tackle more complex geometric problems involving transformations. Understanding dilations and translations not only enhances geometric comprehension but also lays a solid foundation for advanced mathematical studies.