Unit 7 Similarity, Congruence, and Proofs

1. Unit 7 Similarity, Congruence, and Proofs Lesson 1: Investigating Properties of Dialations MCC9-12.G.SRT.1a MCC9-12.G.SRT.1b

2. Investigating properties of parallelism and the center • 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.

3. Introduction • Specifically, we are going to study non-rigid motions of dilations. • Dilationsare transformations in which a figure is either enlarged or reduced by a scale factor in relation to a center point.

4. 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.

5. Use a ratio of corresponding sides to find the scale factor: The scale factor, k, takes a point Pand moves it along a line in relation to the center so that

6. Key Concepts • 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.

7. Key Concepts • 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.

8. Key Concepts • In transformations, the corresponding sides are the preimage and image sides, so AB and A’B’ 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).

9. Key Concepts • 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.

11. Example 1 Is the transformation on the right a dilation? Justify your answer using the properties of dilations.

12. 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.

13. Verify that the corresponding sides are parallel. • By inspection, because both lines are vertical; therefore, they have the same slope and are parallel.

14. In fact, these two segments, and , lie on the same line. • All corresponding sides are parallel.

15. 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. • Verify that the distances of the corresponding sides have changed by a common scale factor, k.

16. Each vertex’s preimage coordinate is multiplied by 2 to create the corresponding image vertex. Therefore, the common scale factor is k = 2. 3. Verify that corresponding vertices are collinear with the center of dilation, C. • 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.

17. 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.

18. Example 2 • The following transformation represents a dilation. What is the scale factor? Does this indicate enlargement, reduction, or congruence?

19. Determine the scale factor. • Start with the ratio of one set of corresponding sides. • The scale factor appears to be .

20. 2. Verify that the other sides maintain the same scale factor. • Therefore, and the scale factor, k, is .

21. 3. 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.

22. Investigating Scale Factors • A figure is dilated if the preimage can be mapped to the image using a scale factor through a center point, usually the origin. • You have been determining if figures have been dilated, but how do you create a dilation? • If the dilation is centered about the origin, use the scale factor and multiplyeach coordinatein the figure by that scale factor. • If a distance is given, multiply the distance by the scale factor.

23. The notation is as follows: Dk(x, y) = (kx, ky). • Multiply each coordinate of the figure by the scale factor when the center is at (0, 0).

24. The lengths of each side in a figure also are multiplied by the scale factor. • If you know the lengths of the preimage figure and the scale factor, you can calculate the lengths of the image by multiplying the preimage lengths by the scale factor. • Remember that the dilation is an enlargement if k > 1, a reduction if 0 < k < 1, and a congruency transformation if k = 1.

25. Example 1 A triangle has vertices G (2, –3), H (–6, 2), and J (0, 4). If the triangle is dilated by a scale factor of 0.5 through center C (0, 0), what are the image vertices? Draw the preimage and image on the coordinate plane.

26. 1. Start with one vertex and multiply each coordinate by the scale factor, k. Dk = (kx, ky) 2. Repeat the process with another vertex. Multiply each coordinate of the vertex by the scale factor. • Repeat the process for the last vertex. Multiply each coordinate of the vertex by the scale factor.

27. 4. List the image vertices. 5. Draw the preimage and image on the coordinate plane.

28. Example 2 • What are the side lengths of with a scale factor of 2.5 given the preimage and image to the right and the information that DE= 1, EF = 9.2, and FD = 8.6?

29. Choose a side to start with and multiply the scale factor (k) by that side length. • Choose a second side and multiply the scale factor by that side length.

30. Choose the last side and multiply the scale factor by that side length. • Label the figure with the side lengths.