STRUCTURE OF MATH II
STRUCTURE OF MATH II MATH 366
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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 366 at Texas A&M University taught by Sandra Nite in Fall. Since its upload, it has received 10 views. For similar materials see /class/226016/math-366-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
Section 121 Math 366 Lecture Notes Section 121 Translations and Rotations Translations Any rigid motion that preserves length or distance is an isometry meaning equal measure Any function from a plane to itself that is a onetoone correspondence between a plane and itself is a transformation of the plane A translation is a motion of a plane that moves every point of the plane a speci ed distance in a speci ed direction along a straight line Properties of Translations o A gure and its image are congruent o The image of a line is a line parallel to it Constructions of Translations Construct the image A of pointA in the direction and magnitude of vector W A M Construct parallelogram MAA N so that M is in the same direction as W 1 Draw ray MA 2 Construct ray AP parallel to MN 3 Construct pointA on E the same distance from A as M is from N Section 121 A geoboard or a grid can be used to find an image of a set of points Find the image of under the translation from X toX in the figure below B n a n o o o o n o u o c n o u a o o o X o u o o o o Coordinate Representation of Translations In the figure below AA 13 C is the image of AABC under the translation defined by the slide arrow from O to 0 where O is the origin and O has coordinates 5 2 The table below shows how the coordinates of the image vertices are obtained from the original vertices Point Image Point ray x 5139 2 A 2 6 C13 Section 1271 Property of a Translation in a Coordinate System A translation is a inction from the plane to the plane such that to every point x y corresponds the point x a y b where a and b are real numbers Notation x y gt x a y b Find the coordinates of the image of the vertices of quadrilateral ABCD in the gure below under the translations given Draw the image in each case xyx2y3 a translation determined by the slide arrow fromAl 2 toA 3 l a 8 D C7 D C 5 5 5 5 A B3 A E 2 2 1 1 HUMqu HHHHH1 BE5 4321112345678 3s54321 12345573 2 2 3 3 A4 4 5 5 5 5 7 7 3 8 A rotation or turn is another kind of isometry The gure below illustmtes congruent gures that resulted from a rotation about point 0 Point O is the turn center and A O is the turn angle F ii A rotation can be constructed with tracing paper Section 121 A rotation is a transformation of the plane determined by holding one point the center xed and rotating the plane about this point by a certain amount in a certain direction Usually a positive rotation is counterclockwise and a negative rotation is clockwise A rotation of 360 about a point will move any point and gure onto itself Such a transformation is an identity transformation A rotation of 180 is a half tum Rotations are useful in determining turn symmetries Construction of a Rotation Construct the image of point P under a rotation with center 0 through the angle and in the direction given in the gure below 39P A 1 1 Construct an isosceles triangle BAC with B on one side of the given angle and C on the other side so thatAB AC OP 2 Construct A POP congruent to A BAC Slopes 0f Perpendicular Lines Two lines neither of which is vertical are perpendicular iff their slopes m1 and m satisfy the condition mlmz 1 Every vertical line has an unde ned slope and is perpendicular to a line with slope 0 Are the lines 2x 7 3y 7 and 3x 7 2y 5 parallel perpendicular or neither Find the equation of the line through 3 2 and perpendicular to the line 3x y 4