In the realm of geometry, dilations offer a fascinating lens through which we can transform and resize figures while preserving their shape. In this comprehensive guide, we delve into the art of dilations, explore their properties, and present a set of practice problems for you to hone your skills. Accompanying these problems is a detailed answer key to reinforce your understanding.
Exploring the World of Dilations
Dilations are a fundamental geometric transformation that involves changing the size of a figure while keeping its shape intact.
Skill 1: Understanding the Dilation Process
Dilations involve a center of dilation and a scale factor. The scale factor determines how much the figure is resized, while the center of dilation serves as the point of reference.
Skill 2: Performing Dilations on Various Figures
Dilations can be applied to a variety of shapes, including polygons and circles. Understanding how dilations affect different types of figures enhances your geometric intuition.
Solving Practice Problems with Dilations
Let’s put our knowledge to the test with practice problems:
- Dilate triangle ABC with vertices A(2, 3), B(4, 1), and C(1, 5) by a scale factor of 2, centered at the origin. Determine the coordinates of the dilated vertices.
Answer: The dilated vertices are A'(4, 6), B'(8, 2), and C'(2, 10).
- Given a circle with center O(0, 0) and radius r = 5, perform a dilation with a scale factor of 3. Find the equation of the dilated circle.
Answer: The equation of the dilated circle is x² + y² = 45.
- Dilate quadrilateral WXYZ with vertices W(1, 1), X(5, 1), Y(4, 5), and Z(2, 4) by a scale factor of 0.5, centered at C(3, 3). Determine the coordinates of the dilated vertices.
Answer: The dilated vertices are W'(2, 2), X'(4, 2), Y'(3, 6), and Z'(1.5, 3.5).
Mastering Geometric Transformations: Navigating Dilations with Precision
As you delve into the world of dilations, you gain the ability to resize and transform figures while maintaining their inherent properties. This skill extends beyond geometry, finding applications in architecture, art, and engineering.
In conclusion, dilations offer a captivating approach to geometric transformations. By understanding the dilation process, performing dilations on various figures, and solving practice problems, you equip yourself to confidently navigate a range of dilation scenarios. Armed with the practice problems and detailed answer key provided in this guide, you’re well-prepared to harness the transformative power of dilations with confidence.