Geometry. Topic 1 Transformations and Congruence

Transcription

1 Geometry Topic 1 Transformations and Congruence

2 MAFS.912.G-CO.1.2 Consider the point A at ( 3, 5). A. Find the coordinates of A, the image of A after the transformation: (, ) (, ). B. What type of transformation is (, ) (, )? C. Find the coordinates of A, the image of A after the transformation: (, ) ( 2, + 4). D. Write a transformation rule to map A to A. A.(3, -5) B. Rotation of 180 C. (1, -1) D.(x, y) (x 4, y + 6)

3 MAFS.912.G-CO.1.3 A regular polygon with sides is carried onto itself by a positive rotation about its center that is a multiple of 60, but less than 360. Which could NOT be the value of? A. 3 B. 4 C. 5 D. 6 D

4 MAFS.912.G-CO.1.3 Use the diagram below of parallelogram to answer the following three questions. A E B H I F D A reflection across carries parallelogram onto itself. True X False A rotation of 90 about I carries parallelogram onto itself. True X False G C Continue on next slide

5 MAFS.912.G-CO.1.3 Use the diagram below of parallelogram to answer the following three questions. A E B H I F A rotation of 180 about I carries parallelogram onto itself. True X False D G C

6 MAFS.912.G-CO.1.5 Given point Ais located at (1, 3). What is the final image of Aafter this series of transformations? Reflect A across the y-axis. Translate the image such that (, ) ( 4, + 2). A. ( 1, 3) B. ( 3, 5) C. ( 3, 1) D. ( 5, 5) D

7 MAFS.912.G-CO.1.3 What type(s) of symmetry does a regular octagon have? Reflection symmetry and rotational symmetry

8 MAFS.912.G-CO.1.3 Consider the figure shown to the right. How many lines of symmetry does the figure have? Explain. The figure does not have line symmetry because there is no line that can be drawn through the figure so that the preimage, after reflecting across the line, is the original image. A. Does the figure have rotational symmetry? If so, describe it. A. Rotational, 180 about the center B. How could you alter the existing figure so that it has two more lines of symmetry than the given figure? Provide a sketch to explain your answer. B. Check student work.

9 MAFS.912.G-CO.1.4 In a drawing that contains points,,, and the origin, = 50 and =80. A. Explain how you know that cannot be the image of under a rotation with center. A. If is the preimage of a rotation with image, then the point has image and point has image. A rotation about point causes all angles with vertex formed by a point and its image to have the same measure. If point has image and point has image, then =. However,, because Thus, cannot be the image of under a rotation with center. B. Explain how you could change the given information so that would be an image of under a rotation with center. B. If =, =, and =, then would be the image of under a rotation with center.

10 MAFS.912.G-CO.1.5 Suppose a point (, ) on the coordinate plane is rotated 270 counterclockwise about the origin, then reflected across the line =, then reflected across the -axis. Show that the image of (, ) after this sequence of transformations is the same as the image of (, ) after a 180 rotation about the origin. (, ) > (, ) > (, ) > (, ) which is equivalent to 180 rotation (, )

11 MAFS.912.G-CO.1.2 A triangle with ver ces ( 3, 1), (1, 5), and (4, 1) is reflected across the axis. What are the vertices of the image? ( 3, 1) (1,-5) (4, 1)

12 MAFS.912.G-CO.1.2 What term describes a transformation that does not change a figure s size or shape? A. Similarity B. Rigid motion C. Collinearity D. Symmetry B

13 MAFS.912.G-CO.1.2 The point (4, 3) is rotated 90 about the origin. In which quadrant is? A. I B. II C. III D. IV D

14 MAFS.912.G-CO.2.6 A rigid motion,, of the plane takes a point,, as input and gives as output (i.e., = ). Similarly, = for input point and output point. Jerry claims that knowing nothing else about, we can be sure that because rigid motions preserve distance. A. Show that Jerry s claim is incorrect by giving a counterexample (hint: a counterexample would be a specific rigid motion and four points,,, and in the plane such that the motion takes to and to, yet ). A. Check student work for counter example. Example 1: using rotation and then the distance formula to find the distance of AC and then BD, it can be shown that AC is not congruent to AD. Example 2: using reflection, it can be shown that the distance from A to the line of reflection is different from the distance of B to the line of reflection thereby the length of AC is not congruent to the length of BD. Continue on next slide

15 MAFS.912.G-CO.2.6 A rigid motion,, of the plane takes a point,, as input and gives as output (i.e., = ). Similarly, = for input point and output point. Jerry claims that knowing nothing else about, we can be sure that because rigid motions preserve distance. B. There is a type of rigid motion for which Jerry s claim is always true. Which type below is it? Rotation Reflection X Translation Continue on next slide

16 MAFS.912.G-CO.2.6 A rigid motion,, of the plane takes a point,, as input and gives as output (i.e., = ). Similarly, = for input point and output point. Jerry claims that knowing nothing else about, we can be sure that because rigid motions preserve distance. C. Suppose Jerry claimed that. Would this be true for any rigid motion that satisfies the conditions described in the first paragraph? Why or why not? C. Yes, because rigid motions always preserve distance.

17 MAFS.912.G-CO.1.5 Use the figure to the right. A. Transform by reflecting it across the to produce. A. (4,6), B (8,4), C (4,4) B. Describe a transformation, or composition of transformations, that maps to. B. Check student work. C. Describe a single transformation that maps to. C. Since y = 2.5 is perpendicular to the y-axis, a rotation of 180 about (0, 2.5) maps ABC to DEF

18 MAFS.912.G-CO.2.6 & MAFS.912.G-CO.2.7 Given right triangles and where and are right angles, and. Describe a precise sequence of rigid motions which would show. Sample Answer 1: Translate ABC along CF such that point C maps onto F, resulting in image A B C then reflect A B C onto DEF. Sample Answer 2: reflect ABC over the perpendicular bisector of EB such that ABC maps onto DEF.

19 MAFS.912.G-CO.1.2 The vertices of have coordinates J(5, 1), K( 2, 3), and L( 4, 1). Under which transformation is J K L not congruent to JKL? A. A translation of two units to the right and two units down. B. A counterclockwise rotation of 180 degrees around the origin. C. A reflection over the x-axis. D. A dilation with a scale factor of 2 and centered at the origin. D

20 MAFS.912.G-CO.1.5 & MAFS.912.G-CO.2.7 Given the figure to the right: A. Reflect triangle ABC about the y-axis and label the new figure triangle A B C. B B. Give a sequence of 2 rigid transformations that would also carry triangle ABC onto triangle A B C. B. Sample Answer: Reflect the over = 2, then translate two units to the right. A C Continue on next slide

21 MAFS.912.G-CO.1.5 & MAFS.912.G-CO.2.7 C. Find the side lengths and angle measures of both triangles. Show your work, and explain your reasoning. C.,,, = 4 2, = 8 =90,,, =45, D. What do you know about the two triangles? Explain your reasoning. D. Sample Answer: due to Sideside-side postulate. B A C Continue on next slide

22 MAFS.912.G-CO.1.5 & MAFS.912.G-CO.2.7 E. Draw a new triangle PQR that is congruent to triangle ABC, and in a different location than either triangle ABC or triangle A B C. Explain how you know triangle PQR is congruent to triangle ABC using at least 2 transformations. E. Check student work. A B C

23 MAFS.912.G-CO.1.2 What are the coordinates of, the image of (2,4), after a counterclockwise rotation of 90 about the origin? A. ( 2,4) B. ( 2, 4) C. ( 4,2) D. ( 4, 2) C

24 MAFS.912.G-CO.2.7 As graphed on the set of axes below, is the image of after a sequence of transformations. Is congruent to? Use the properties of rigid motion to explain your answer. Sample Answer: Yes, because following two rigid transformations are used. Reflect over the Translate three units down

25 MAFS.912.G-CO.1.5 Each of the illustrations below shows in black a plane figure consisting of the letters F, R, E, and D evenly spaced and arranged in a row. In each illustration, an alteration of the black figure is shown in gray. In some of the illustrations, the gray figure is obtained from the black figure by a geometric transformation consisting of a single rotation. In others, this is not the case. Continue on next slide

26 MAFS.912.G-CO.1.5 A. Which illustrations show a single rotation? A. Figures 2 and 5 B. Some of the illustrations are not rotations or even a sequence of rigid transformations. Select one such illustration, and use it to explain why it is not a sequence of rigid transformations. B. Sample Answer: Figure 1, shows translation but each letter is translated a different distance. Since translation requires a shift of the entire plane by the same distance, the figure does not qualify.