Introduction
Unit transformations are a fundamental concept in mathematics, enabling us to manipulate geometric shapes on a coordinate plane. Homework 2 in this unit delves deeper into the world of transformations, challenging students to apply their knowledge of translation, reflection, rotation, and dilation to solve complex problems. In this article, we present an all-encompassing answer key to help students grasp these concepts and confidently conquer the exercises.
Overview of Transformations
Transformations are processes that alter the position, orientation, or size of a shape while preserving its inherent properties. The four primary transformations are translation, reflection, rotation, and dilation.
Homework 2 Exercise 1: Translation
Translation involves shifting a shape’s position without changing its size or shape. To perform a translation, we need to add the same value to each corresponding coordinate of the shape.
Example:
Translate quadrilateral ABCD four units to the right and three units upward. The original coordinates of ABCD are A(2, 5), B(4, 8), C(7, 6), and D(5, 3).
Answer:
A’: (6, 8), B’: (8, 11), C’: (11, 9), D’: (9, 6)
Homework 2 Exercise 2: Reflection
Reflection creates a mirror image of a shape across a specified line, known as the line of reflection. Each point on the original shape corresponds to a point on the reflected shape, equidistant from the line of reflection.
Example:
Reflect triangle XYZ over the y-axis. The original coordinates of XYZ are X(3, 2), Y(6, 4), and Z(4, 7).
Answer:
X’: (-3, 2), Y’: (-6, 4), Z’: (-4, 7)
Homework 2 Exercise 3: Rotation
Rotation involves turning a shape around a fixed point known as the center of rotation. The direction and angle of rotation determine the extent of the turn.
Example:
Rotate pentagon PQRST 90 degrees counterclockwise around the origin. The original coordinates of PQRST are P(2, 1), Q(3, 3), R(5, 2), S(4, 0), and T(6, 0).
Answer:
P’: (-1, 2), Q’: (-3, 3), R’: (-2, 5), S’: (0, 4), T’: (0, 6)
Homework 2 Exercise 4: Dilation
Dilation resizes a shape by a scale factor without changing its shape. The scale factor can be greater than 1 for an enlargement or between 0 and 1 for a reduction.
Example:
Dilate circle O with a scale factor of 1.5, centered at the origin. The original coordinates of points on the circle are A(2, 3), B(4, 6), and C(6, 2).
Answer:
A’: (3, 4.5), B’: (6, 9), C’: (9, 3)
Homework 2 Exercise 5: Composite Transformations
Composite transformations involve combining two or more transformations to achieve a specific result.
Example:
Perform a translation of triangle LMN six units to the right and a dilation with a scale factor of 2, centered at the origin. The original coordinates of LMN are L(1, 3), M(4, 3), and N(2, 1).
Answer:
L’: (7, 3), M’: (16, 3), N’: (12, 1)
Conclusion
Unit transformations play a crucial role in understanding geometry and its applications in real-life scenarios. Homework 2 presents students with challenging exercises that cover translation, reflection, rotation, and dilation. This comprehensive answer key serves as a valuable resource for students to verify their solutions, reinforce their understanding, and approach similar problems with confidence. Emphasize the importance of practice and exploration to further strengthen their grasp on transformations. With dedication and perseverance, mastering these concepts will unlock new possibilities in the realm of mathematics and beyond. Happy learning!
