11/24/2023 0 Comments Rigid motion![]() ![]() Students are introduced to the concept of a construction, and use the properties of circles to construct basic geometric figures. Unit 1 begins with students identifying important components to define- emphasizing precision of language and notation as well as appropriate use of tools to represent geometric figures. This unit lays the groundwork for constructing mathematical arguments through proof and use of precise mathematical vocabulary to express relationships. These rigid motion transformations are introduced through points and line segments in this unit, and provide the foundation for rigid motion and congruence of two-dimensional figures in Unit 2. Transformations that preserve angle measure and distance are verified through constructions and practiced on and off the coordinate plane. The four most common reflections are performed over the following lines of reflection: the $x$-axis, the $y$-axis, $y =x$, and $y =-x$.In Unit 1, students are introduced to the concept that figures can be created by just using a compass and straightedge using the properties of circles, and by doing so, properties of these figures are revealed. However, the orientation of the points or vertices changes when reflecting an object over a line of reflection. In fact, in reflection, the angle measures of the objects, parallelism, and side lengths will remain intact. The distances between the vertices of the triangles from the line of reflection will always be the same. The graph above showcases how a pre-image, $\Delta ABC$, is reflected over the horizontal line of reflection $y = 4$. This makes reflection a rigid transformation. When learning about point and triangle reflection, it has been established that when reflecting a pre-image, the resulting image changes position but retains its shape and size. In reflection, the position of the points or object changes with reference to the line of reflection. Once we’ve established their foundations, it will be easier to work on more complex examples of rigid transformations. We’ll explore different examples of reflection, translation and rotation as rigid transformations. It’s time to explore these three examples of basic rigid transformations first. This makes this transformation a rigid transformation. Rotation: In rotation, the pre-image is “turned” about a given angle and with respect to a reference point, retaining its original shape and size.The image is the result of “sliding” the pre-image but its size and shape remain the same. Translation: This transformation is a good example of a rigid transformation.Reflection: This transformation highlights the changes in the object’s position but its shape and size remain intact.These three transformations are the most basic rigid transformations there are: Some examples of rigid transformations occur when a pre-image is translated, reflected, rotated or a combination of these three. This is why it’s essential to have a refresher and understand why they’re each classified as a rigid transformation. This shows that when dealing with rigid transformations, it is important to be familiar with the three basic rigid transformations. ![]() The series of basic rigid transformations still result in a more complex rigid transformation. The reflected square is then translated $10$ units to the right and $20$ units downward.The reflected points are $5$ units from the left of the vertical line $x = -5$. The square $ABCD$ is reflected over the line $x = -5$.Breaking down the series of transformations performed on the pre-image highlights the story behind the rigid transformation: This shows that the transformation performed on the square is a rigid transformation. Read more How to Find the Volume of the Composite Solid? ![]()
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