UNIT 1 - tRANSFORMATIONS IN THE cOORDINATE PLANE



41685913447UNIT 1 - tRANSFORMATIONS IN THE cOORDINATE PLANE00UNIT 1 - tRANSFORMATIONS IN THE cOORDINATE PLANEcenter4237841NAME: ___________________________________PERIOD: ______2018 – 2019 GeometryPebblebrook High School - Deuire00NAME: ___________________________________PERIOD: ______2018 – 2019 GeometryPebblebrook High School - DeuireVocabulary Buildercenter29605900Choose the word from the list below that best matches each phrase.1. The figure that results from a transformation __________________2. The original figure in a transformation ________________________3. Flipping, sliding, or turning a figure __________________________4. Two or more transformations in combination _____________________ ___ __________________________5. This transformation is an example of a _________________________ because the figure slides in one direction, but does not flip, turn, or change size. 41947354258006. This translation is an example of a (n) _________________________ because it preserves distance and angle measures. right17108007. In a translation, the sides or angles of the preimage and image that have the same lengths or angle measures are _________________________.right22232500Use words from the list below to complete the sentences. 990600935841008. In the coordinate plane above, all the triangles are ________________________ figures. 9. To show that any two figures above are congruent, you can identify a __________________ ___________________________ that maps one figure to another.10. A transformation that maps ?ABC to ?DEF is a _______________________ that slides ?ABC four units to the right and two units down. 11. A transformation that maps ?DEF to ?GHI is a ______________________ with _____________ ____ ___________________ of 180° and ____________ _____ ________________________ at the origin.12. A transformation that maps ?GHI to ?JKL is a _________________________ with ________ ____ ______________________ of x=4.13. A transformation that maps ?ABC to ?JKL is a ____________ _________________ by sliding ?ABC twelve units to the right and two units down and then reflecting across the x-axis. 1-1 Precise Definitions Vocabulary WordDefinitionExampleLine SegmentAngleCircleRadiusParallel LinesPerpendicular LinesSegment BisectorMidpointAngle BisectorAdjacent AnglesVertical AnglesComplementary AnglesSupplementary AnglesCollinearVertexRay-56417930737800Classifying AnglesExample 1Classify each angle as acute, obtuse, or right.14119405499000left3354300Example 2AC⊥FE, AE⊥GC and C is the midpoint of AE. Determine whether each of the following is true or false.-18825923282100368449313869000Example 347138291180300Identify each pair of angles as adjacent, vertical, complementary, or supplementary.2455022189939b.00b.-179929173019a.00a.2419985184860d.00d.-128532173990c.00c. ∠1 and ∠5 ∠3 and ∠4 Find the value of x.right13066100The sum of the measure of a triangle is 1800.Example 4center26953900Find the value of x.240702344375300When you extend the sides of a polygon, the original angles may be called the interior angles and the angles that form linear pairs with the interior are the exterior angles.Each exterior angle of a triangle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. *The sum of the 2 remote interior angles equal the exterior angle.144612013709Example 5Find the value of x.center190127002862543127002.2.578485122891.1.A(n) isosceles triangle has two congruent sides called the legs. The angle formed by the legs is called the vertex angle. The other two angles are called base angles.The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.163785214578800Example 6Find the value of each variable.-457200151354001-2 TranslationsA transformation is a change in the position, shape, or size of a figure. The original figure is called the pre-image. The resulting figure is called the image. A transformation maps the preimage to the image. Arrow notation (→) is sometimes used to describe a transformation, and primes ( ' ) are used to label the image. A translation is an operation that slides a geometric figure in the plane. center2388348right4152228-228600389255Translations are a ______________________, or _________________.Translations are ____________________________, and preserve _______________________________.Translations can be achieved by performing two composite ___________________________ over _________________________ lines.00Translations are a ______________________, or _________________.Translations are ____________________________, and preserve _______________________________.Translations can be achieved by performing two composite ___________________________ over _________________________ lines.In a horizontal translation, the x-coordinate changes but the y-coordinate stays the same. This translation can by represented by the function Tx, y=(x+a, y)41412838355900377190022733000In a vertical translation, the y-coordinate changes, but the x-coordinate stays the same.This translation can by represented by the function Tx, y=(x, y+b)In a slant translation, both the x- and y-coordinate change.46733283547000This translation can by representedby the function Tx, y=(x+a, y+b)Example 1Find the coordinates for the image of ?ABC after the translation (x, y)→(x+2, y-1). Draw the image. -21492918579400A’ ( , )B’ ( , )C’ ( , )Example 2A four-sided figure has the following coordinates A(1, -5), B(2, -6), C(3, -7), D(4, -8). After a translation, its coordinates are A'(6, -7), B'(7, -8), C'(8, -9), D'(9, -10). Write the rule for the translation. Example 3R’ falls on the point (1, -2) after the transformation of T(x, y)=(x+4, y-6). What is the coordinate of R?1-3 Reflections A reflection is a transformation that flips a figure across a line called a line of reflection. -217805328930Reflections are a ________________.The flip is performed over a _________ ____ _____________________.Reflections are ______________________, but DO NOT preserve _________________________________.00Reflections are a ________________.The flip is performed over a _________ ____ _____________________.Reflections are ______________________, but DO NOT preserve _________________________________.4058194-73306300When a point is reflected across the y-axis, the sign of its x-coordinate changes. The function for a reflection across the y-axis is Ry-axisx,y=(-x, y)3991684-63948300When a point is reflected across the x-axis, the sign of its y-coordinate changes. The function for a reflection across they-axis is Rx-axisx,y=(x, -y)49888602099200Another common line of reflection is the diagonal line y=x. To reflect over this line, swap the x- and y-coordinates.The function for a reflection across line y=x is Ry=xx,y=(y, x)To reflect over the line y =-x, swap and opposite sign the x- and y-coordinates.403281010331900The function for a reflection across line y=-x is Ry=-xx,y=(-y, -x)373783424055300To reflect through the origin, the sign of the x and y coordinates change.The function for a reflection through the origin is R(0,0)x,y=(-x, -y) Example 1Reflect the figure with the given vertices across the given line. 1882588814300Reflect over the x-axisX (2, -1)Y (-4, -3)Z (3, 2)27566475767300Reflect over the y-axisS (3, 4)T (3, 1)U (-2, 1)V(-2, 4)184168722169000Reflect over the y=xR (-2, 2)S (5, 0)T (3, -1)To reflect over the line x = a (where a is a number on the x-axis), multiply a times 2 and subtract x for the x-coordinate and leave y the same for the y-coordinate.The function for a reflection across line x=a is Rx=ax,y=(2a-x, y)To reflect over the line y = b (where b is a number on the y-axis), leave x the same for the x-coordinate and multiply b times 2 and subtract for the y-coordinate.The function for a reflection across line y=b is Ry=bx,y=(x, 2b-y)Example 2237923325852000Reflect the figure with the given vertices across the given line. Reflect over the y=-1A (-1, 1)B (-5, 1)C (-4, 2)D (-2, 2)2258396-22818900Reflect over x=2P (4, 2) Q (3, 0)R (5, -5)215106725355100 Reflect over the y=-xD (1, 1)E (3, 2)F (2, 4)1-4 Rotationscenter39541800A rotation is a transformation that turns a figure around a point, called the center of rotation. Counter-clockwise is considered the positive direction, so clockwise is considered the negative direction. -1161144173 00 A 90° rotation is equivalent to a 270o CWrotation and has the function:R90°x, y=(-y, x)A 180° rotation is equivalent to a 180o CW rotation and has the function:R180°x, y=(-x, -y)A 270° rotation is equivalent to a 90o CW rotation and has the function:R270°x, y=(y, -x)3141171-3490000Example 1755856122947(x, y)→(-x,-y)00(x, y)→(-x,-y)3361845720180° CW00180° CW22456601698440091440115288600-10757616984400306452513673300-1860868388416Rotation in the Coordinate Plane00Rotation in the Coordinate Plane772984227590(x, y)→(y,-x)00(x, y)→(y,-x)1944245746790° CW0090° CW23263411661100011161051392140012894111213300-18826020245300301124212988000612813139812(x, y)→(-y,x)00(x, y)→(-y,x)2327212197690° CCW0090° CCW224536095250010623171019700Example 2Triangle GHJ is graphed on the coordinate plane. Draw the image of this triangle after counterclockwise rotations of 90°, 180°, and 270° about the origin.-539757661100-99463437136290o CCW0090o CCW-8101815949700-1009053236406180o CCW00180o CCW-9412924275700-180415227218270o CCW00270o CCW1-5 SymmetryA regular polygon is a polygon with all sides equal in length and all angles equal in measure. If a regular polygon has n sides, then it also has n lines of symmetry.When you reflect a figure over line of symmetry, the image is congruent to and in the same location as the original pre-image. When this happens, we say that the reflection maps the figure onto itself. -11584336975100This type of symmetry is called line symmetry or reflection symmetry. 31196648545300Example 1Tell whether each figure has line symmetry. 559136138210034552961176600a) b) 150894264080500A figure that has rotational symmetry will map onto itself more than once during a 360° turn. To find the rotational symmetry, divide 360o by the number of sides. Example 2Determine whether each figure has rotational symmetry. If so, describe the rotation that maps the figure onto itself. 388620013470700295835649800a) b)Example 3List all the transformations that map the following graphs onto itself.376457917892002420466574100a) b)Line Symmetry - Line Symmetry – Rotational Symmetry - Rotational Symmetry - 1-6 Sequence of Transformations Sometimes, more than one transformation is needed to produce a particular image from a given pre-image. To determine the necessary sequence of transformations, compare the image to the pre-image. If the orientation of the figure has changed, then a rotation or reflection has probably taken place. A composition of transformations is one transformation followed by another. A glide reflection is the composition of a translation and a reflection across a line parallel to the vector of translation. right12700Example 1Draw the result of each composition of transformations. Reflect the triangle over the liney=1, then translate 3 units down. left1456700Example 2Draw the result of each composition of transformations.Reflect the triangle over the x-axis, then translate 3 units to the left. left15949700Example 3Identify a sequence of transformations that will map each pre-image onto its final image. Use correct transformation notation. left5617900Example 4Identify a composition of transformations that will map each pre-image onto its final image. Use correct transformation notation.center34822900 ................
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