GEOMETRY MODULE 2 LESSON 6 DILATIONS AS TRANSFORMATIONS OF THE PLANE LESSON OBJECTIVE • Review the properties of basic rigid motions. • Understand the properties of dilations and that a dilation is also an example of a transformation of the plane. CLASSWORK Recall, the properties of any rigid transformation. • Rigid motions map a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. • Rigid motions preserve lengths of segments and measures of angles. 1. Find the center and the angle of rotation that takes �� to �′�′. Find the image P’ of point P under this rotation. Step 1: Determine the location of center O as the intersection of the perpendicular bisectors of ��′ and �′�′. Step 2: Determine the angle of rotation by joining A to O and O to �’, then measuring the angle CCW. MOD2 L6 1 2. Find the line of reflection for the reflection that takes point A to point �’. Find the image �’ under this reflection. 3. Quinn tells you that the vertices of the image of quadrilateral CDEF reflected over the line � = − ! � + 2 are the following: �! −2, 3 , �! 0,0 , �! −3, −3 , and �!(−3, 3). Do you ! agree with Quinn? Explain. The given coordinates for the reflection are incorrect. MOD2 L6 2 4. The point �’ is the image of point C under a translation of the plane along a vector. Find the coordinates of C if the vector used for the translation is ��. The coordinates for C if the vector �� is used are (−5, 4). DISCUSSION: DILATIONS VS RIGID MOTIONS • What do dilations have in common with translations, reflections, and rotations? Each transformation follows a rule assignment. Dilations and rotations require a center. Dilations and rigid motions preserve angle measures. • What is different about dilations compared to translations, reflections, and rotations? Dilations do not preserve distance (side lengths) while rigid motions do. • An additional feature that dilations share with rigid motions is the existence of an inverse dilation, just as inverses exist for the rigid transformations. What this means is that the composition of a dilation and its inverse takes every point in the plane back to itself. For example, consider a 90° clockwise rotation about center O of point P. What inverse motion would take P back to itself? 90° counter-clockwise rotation • What would an inverse dilation rely on to bring the image of a dilated point �’ back to P? The reciprocal of the original scale factor PRACTICE MOD2 L6 3 1. A dilation with center O and scale factor r takes A to �’ and B to �’. Find the center O, and estimate the scale factor r. Scale factor is approximately 2. ! 2. Given a center O, scale factor � = 3, and points A and B, find the points � = �!,! � and ! � = �!,! � . Compare the length of AB with the length of �′�′ by division, in other words, find !!!!. How does this number compare to r? !" This measurement is an estimate. �′�′ 12.6 = = 3 = � �� 4.2 SUMMARY • There are two major classes of transformations: those that preserve distance (translations, reflections, rotations) and those that do not (dilations). • Like rigid motions, dilations involve a rule assignment for each point in the plane and also have inverse functions that return each dilated point back to itself. MOD2 L6 4 .