Introduction Think about resizing a window on your computer screen. You can stretch it vertically, horizontally, or at the corner so that it stretches both horizontally and vertically at the same time. These are non-rigid motions. Non-rigid motions are transformations done to a figure that change the figure’s shape and/or size. These are in contrast to rigid motions, which are transformations to a figure that maintain the figure’s shape and size, or its segment lengths and angle measures. 1.1.1: Investigating Properties of Parallelism and the Center
Introduction, continued Specifically, we are going to study non-rigid motions of dilations. Dilations are transformations in which a figure is either enlarged or reduced by a scale factor in relation to a center point. 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts Dilations require a center of dilation and a scale factor. The center of dilation is the point about which all points are stretched or compressed. The scale factor of a figure is a multiple of the lengths of the sides from one figure to the transformed figure. Side lengths are changed according to the scale factor, k. The scale factor can be found by finding the distances of the sides of the preimage in relation to the image. 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued Use a ratio of corresponding sides to find the scale factor: The scale factor, k, takes a point P and moves it along a line in relation to the center so that . 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued If the scale factor is greater than 1, the figure is stretched or made larger and is called an enlargement. (A transformation in which a figure becomes larger is also called a stretch.) If the scale factor is between 0 and 1, the figure is compressed or made smaller and is called a reduction. (A transformation in which a figure becomes smaller is also called a compression.) If the scale factor is equal to 1, the preimage and image are congruent. This is called a congruency transformation. 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued Angle measures are preserved in dilations. The orientation is also preserved. The sides of the preimage are parallel to the corresponding sides of the image. The corresponding sides are the sides of two figures that lie in the same position relative to the figures. 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on. The notation of a dilation in the coordinate plane is given by Dk(x, y) = (kx, ky). The scale factor is multiplied by each coordinate in the ordered pair. The center of dilation is usually the origin, (0, 0). 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued If a segment of the figure being dilated passes through the center of dilation, then the image segment will lie on the same line as the preimage segment. All other segments of the image will be parallel to the corresponding preimage segments. The corresponding points in the preimage and image are collinear points, meaning they lie on the same line, with the center of dilation. 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued 1.1.1: Investigating Properties of Parallelism and the Center
Key Concepts, continued Properties of Dilations Shape, orientation, and angles are preserved. All sides change by a single scale factor, k. The corresponding preimage and image sides are parallel. The corresponding points of the figure are collinear with the center of dilation. 1.1.1: Investigating Properties of Parallelism and the Center
Common Errors/Misconceptions forgetting to check the ratio of all sides from the image to the preimage in determining if a dilation has occurred inconsistently setting up the ratio of the side lengths confusing enlargements with reductions and vice versa 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice Example 1 Is the transformation on the right a dilation? Justify your answer using the properties of dilations. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued Verify that shape, orientation, and angles have been preserved from the preimage to the image. Both figures are triangles in the same orientation. The angle measures have been preserved. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued Verify that the corresponding sides are parallel. By inspection, because both lines are vertical; therefore, they have the same slope and are parallel. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued In fact, these two segments, and , lie on the same line. All corresponding sides are parallel. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued Verify that the distances of the corresponding sides have changed by a common scale factor, k. We could calculate the distances of each side, but that would take a lot of time. Instead, examine the coordinates and determine if the coordinates of the vertices have changed by a common scale factor. The notation of a dilation in the coordinate plane is given by Dk(x, y) = (kx, ky). Divide the coordinates of each vertex to determine if there is a common scale factor. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued Each vertex’s preimage coordinate is multiplied by 2 to create the corresponding image vertex. Therefore, the common scale factor is k = 2. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued Verify that corresponding vertices are collinear with the center of dilation, C. 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued A straight line can be drawn connecting the center with the corresponding vertices. This means that the corresponding vertices are collinear with the center of dilation. 1.1.1: Investigating Properties of Parallelism and the Center
✔ Guided Practice: Example 1, continued Draw conclusions. The transformation is a dilation because the shape, orientation, and angle measures have been preserved. Additionally, the size has changed by a scale factor of 2. All corresponding sides are parallel, and the corresponding vertices are collinear with the center of dilation. ✔ 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 1, continued http://www.walch.com/ei/00129 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice Example 3 The following transformation represents a dilation. What is the scale factor? Does this indicate enlargement, reduction, or congruence? 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 3, continued Determine the scale factor. Start with the ratio of one set of corresponding sides. The scale factor appears to be . 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 3, continued Verify that the other sides maintain the same scale factor. Therefore, and the scale factor, k, is . 1.1.1: Investigating Properties of Parallelism and the Center
✔ Guided Practice: Example 3, continued Determine the type of dilation that has occurred. If k > 1, then the dilation is an enlargement. If 0 < k < 1, then the dilation is a reduction. If k = 1, then the dilation is a congruency transformation. Since , k is between 0 and 1, or 0 < k < 1. The dilation is a reduction. ✔ 1.1.1: Investigating Properties of Parallelism and the Center
Guided Practice: Example 3, continued http://www.walch.com/ei/00130 1.1.1: Investigating Properties of Parallelism and the Center