Introduction
Unit 9 Transformations is a critical topic in mathematics that explores the various ways in which geometric figures can be altered on a coordinate plane. Homework 1 in this unit challenges students to apply their understanding of transformations such as translation, reflection, rotation, and dilation to solve problems involving shapes and points. In this article, we present a comprehensive answer key to help students gain a clear understanding of the concepts and confidently tackle the exercises.
Translation
Translation involves moving a shape from one position to another without changing its size or shape. The translation of a shape is determined by its horizontal (x-axis) and vertical (y-axis) shifts. To perform a translation, add the same value to each corresponding coordinate of the shape.
Example:
Translate triangle ABC six units to the right and four units upward. The original coordinates of ABC are A(2, 3), B(4, 6), and C(6, 3).
Answer:
A'(8, 7), B'(10, 10), C'(12, 7)
Reflection
Reflection is a transformation that creates a mirror image of a shape across a given line called the line of reflection. Each point on the original shape has a corresponding point on the reflected shape, equidistant from the line of reflection.
Example:
Reflect quadrilateral PQRS over the x-axis. The original coordinates of PQRS are P(3, 5), Q(7, 6), R(6, 3), and S(4, 2).
Answer:
P’ (3, -5), Q’ (7, -6), R’ (6, -3), S’ (4, -2)
Rotation
Rotation involves rotating a shape about a fixed point called the center of rotation. The direction of rotation can be clockwise or counterclockwise, and the angle of rotation determines the degree of turning.
Example:
Rotate triangle XYZ 90 degrees counterclockwise around the origin. The original coordinates of XYZ are X(3, 1), Y(2, 4), and Z(5, 5).
Answer:
X’ (-1, 3), Y’ (-4, 2), Z’ (-5, 5)
Dilation
Dilation is a transformation that changes the size of a shape. The scale factor determines whether the dilation is an enlargement (scale factor > 1) or a reduction (scale factor < 1). To perform a dilation, multiply the coordinates of each point by the scale factor.
Example:
Dilate triangle LMN with a scale factor of 2, centered at the origin. The original coordinates of LMN are L(2, 4), M(6, 6), and N(4, 2).
Answer:
L'(4, 8), M'(12, 12), N'(8, 4)
Composite Transformations
Composite transformations involve combining two or more transformations to create a new figure.
Example:
Perform a reflection over the y-axis on triangle PQR (P(2, 3), Q(5, 6), R(7, 2)), and then translate the reflected image five units to the left and two units upward.
Answer:
P” (-7, 5), Q” (-10, 8), R” (-12, 4)
Invariant Points
Invariant points are points that remain unchanged after a transformation. For example, the center of rotation in a rotation or the points lying on the line of reflection in a reflection are invariant points.
Example:
Identify the invariant point(s) in a reflection of pentagon ABCDE over the x-axis.
Answer:
The invariant point(s) are the vertices on the x-axis: A and E.
Conclusion
Unit 9 Transformations Homework 1 introduces students to the fascinating world of geometric transformations. By understanding the principles of translation, reflection, rotation, and dilation, students can explore how shapes and figures can be manipulated on a coordinate plane. This answer key serves as a valuable resource for students to verify their solutions, solidify their understanding, and approach similar problems with confidence. Remember that practice is key to mastering these concepts, so keep practicing and exploring the world of transformations to sharpen your mathematical skills. Happy learning!
