Homework for Section 5.1 1. reate the rotation R(T) 2. reate the reflection F(T) of the triangle T shown below 90 degrees of the triangle T shown below across clockwise about the center point of rotation. the mirror line shown. A D A D B B 3. Translate triangle DEF by vector DD. 4. Sketch the image of the flag after a 270 degree clockwise rotation around point. E D F D
5. Perform the rigid motions indicated below. Reflect the triangle across the mirror line shown. Rotate the triangle 180 degrees around point. 6. For each pair of figures below, find a rigid motion that moves one on top of the other. If the answer is a rotation, locate the center and state the amount of rotation. If the answer is a reflection, sketch the mirror line. If the answer is a translation, give the number of units moved in each direction.
7. Use a rotation, reflection, or translation to show that each pair of figures is congruent. If you use a rotation, locate precisely the center and sketch the angle of rotation. If the answer is a reflection, locate precisely the mirror line. If the answer is a translation, sketch a vector of the translation. 8. In the dot paper at right, find both a rotation and a reflection that will move one triangle onto the other. Locate the center of the rotation and label it. How many degrees is the rotation? Locate the mirror line of the reflection, and draw it in on the dot paper. 9. In the dot paper at right, find both a rotation and a reflection that will move one triangle onto the other. Locate the center of the rotation and label it. How many degrees is the rotation? Locate the mirror line of the reflection, and draw it in on the dot paper.
10. In the coordinate grid at right: (a) Reflect triangle AB across the x-axis to obtain a triangle A B. 4 (b) Reflect triangle A B across the y-axis to obtain a triangle A B. 2 A B (c) What specific single rigid motion moves triangle AB onto triangle A B? Describe it completely. That is: if it is a rotation give the coordinates of the center and the amount of rotation. If it is a translation give the vector of the translation in (x, y) form. If it is a reflection give the equation of the mirror line. -5 5-2 -4 11. In the coordinate grid at right: (a) Sketch the line x = -2 and reflect triangle AB across it to obtain a triangle A B. (b) Sketch the line x = 2 and reflect triangle A B across it to obtain a triangle A B. 6 4 2 (c) There is a translation that moves triangle AB onto triangle A B Give the translation vector, and compare the distance moved by this translation to the distance between the two mirror lines. -5 5-2 -4
12. Based on your work on the previous problems and classwork, make inferences to fill in the blanks below. The first and last blanks should be filled in with the name of a type of rigid motion. If two reflections have mirror lines that intersect, then the composition of the two reflections is a. Moreover, the center of this resulting rigid motion is the of the mirror lines. If two reflections have mirror lines that are parallel, then the composition of the two reflections is a and the distance moved is times the distance moved by the composition. 13. In the coordinate system below, let T be rotation 90 degrees counterclockwise about the origin, and let S be reflection across the horizontal x - axis. If P is the triangle shown, with vertices (1, 2), (1, 3), and (4, 2), determine and plot the triangles T(P) and S(T(P)). 4 2 P -5 5-2 -4 What single rigid motion S T will move P onto S(T(P))? Describe it completely. That is: if it is a rotation give the coordinates of the center and the amount of rotation. If it is a translation give the vector of the translation in (x, y) form. If it is a reflection give the equation of the mirror line.
14. Shown below are two congruent equilateral triangles. (a) Show that the second is a translation image of the first by showing the vector of such a translation. (b) Show that the second is also a rotation image of the first by locating the center and amount of rotation of such a rotation.