Form 2 Mathematics – GEOMETRICAL TRANSFORMATIONS msomimaktaba, November 13, 2018August 17, 2024 GEOMETRICAL TRANSFORMATIONS-A transformation changes the position, size, direction or shape of objects. -Transformation in a plane is a mapping which moves an object from one position to another within the plane. The new position after a transformation is called an imageExamples of transformations areReflectionRotationEnlargement andTranslationSuppose a point p[x, y] in the xy plane moves to a point p΄ [x΄, y΄] by a transformation TP is said to be mapped to P΄ by T and may be indicated asA transformation in which the size of the image is equal to the size of the object is called an Isometric mappingREFLECTION-Reflection is an example of an isometric mapping-Isometric mapping means the distance from the mirror to an object is the same as that from the mirror to the image.-The plane mirror is the line of symmetry between the object and the image. -The line joining the object and the image is perpendicular to the mirror.NOTE -The symbol/letter for reflection is M. -The reflection in X- axis and Y- axis are indicated as Mx and My respectively. -The reflections in lines with certain equations are indicated with their equations as subscriptsFor example: My = x,is given by My = –xA) Reflection in the x-axisExample1. Find the image of the point A(2,1) after a reflection in the x-axisSolution:2(B)Reflection in y-axisFind the image of 0(3,4) under the reflection in the Y-axisSolution:Exercise 1Find the image of the point D(4,2) under a reflection in the x-axisSolution:2.Find the image of the point P(-2,5) under the reflection in the x-axisSolution:Point Q (-4,3) is reflected in the Y- axisSolution:Point R (6, 5) is reflected in the X-axis.Find the coordinates of its image The vertices of a triangle PQR are P (6, 2), Q (–2, 8), R (–5, –1). If triangle PQR is reflected in the Y axis, find coordinates of the vertices of its image.The vertices of rectangle area A (2,3), B (2,-4), C (4, -4), D (4,3) rectangle ABCD is reflected in the Y-axis(a) Find the coordinates of the vertices of its image(b) Draw a sketch to show the imageSolution6(a)The coordinate of the image is A'(-2,3), B'(-2,-4). C'(-4,-4) and D'(-4,3)C) THE REFLECTION IN THE LINE Y = XThe line y = x makes an angle 45º with the x and y axesSee the diagram∴My=x (x,y)=(y,x)ExampleFind the image of point A(1,2) after a reflection in the line y=xD) REFLECTION IN THE LINE Y = -X ∴ My=-x(X,Y)=(-y,-x)ExampleFind the image of B (-3, 4) after a reflection in the line y=-x followed by another reflection in the line y=0Solution The reflection of B (–3, 4) in the line y = –x is B’ (–4, 3) and the image ofB’ (–4, 3) after reflection in the line y = 0 is B’ (–4, –3)NOTE: If P is the object the reflection of point P(x,y) will be: 1. M x-axis P (x,y) = P′ (x, -y) 2. M y-axis P (x,y) = P′ (-x, y) 3. M Y=x P (x,y) = P′ (y,x) 4.M y=-x P (x,y) = P′ (-y, -x) Rotation– -Rotation is a transformation which moves a point through a given angle. -The angle turned through can be either in clockwise or anticlockwise direction.– -Rotation is an isometric mapping and usually denoted as R. Rθ means a rotation through an angle θ– -In the XY plane when θ is measured in the clockwise direction, the angle is -ve and when measured anticlockwise direction the angle is +veExampleFind the image of the point P(1,0) after a rotation through 900 about the origin in anti-clockwise directionTRANSLATION– -Translation is a straight movement without turning.– -A translation is usually denoted by T. For example T(1,1) = (6,1) means that the point (1,1) has been moved to (6, 1) by a translation T.– – This translation will move the origin (0,0) to (5,0) and it is written as T = (5/0).–Examples:A translation takes the origin to (–2, –5) find when it takes (–2, –3) SolutionT (2,–3) = (0, –8)Find the image of the point (1,2) under a rotation through 1800 anti-clockwise about the origin SolutionFind the rotation of the point (6, 0) under a rotation through 900 clockwise about the origin SolutionFind the image of (1,2) after a rotation of –900 ant –clock wise SolutionFind the image of (–3, 5) after a rotation of –1800 Solution The vertices of rectangle PQRS are P(0,0), Q (3,0), R (3,2), S (0, 2). The rectangle is rotated through 900 clockwise about the origin.(a) Find the co-ordinates of its image(b) Draw the imageMore examples on translationTranslation takes the origin to (-2, 5)Find where it takes(a) (-6, 6)(b) (5, 4 )Solution(a)= == === + =: The translation takes (-6,6) to(b) = += : The translation takes (5,4) to (3,9)A translation takes every point a distance of 1 unit to the left and 2 units downwards on the xy-plane.Find where it takes(a) (0,0)(b) (1,1)(c) (3,7)Solution(a).= += +=: . The translation takes the origin to (–1, –2) (b). = += +=3. A translation moves the origin a distance 2 units along the line y= x upwards.Find where it takes(a) (0,0)(b) (2, –1)(c) (1, 1)Solutionx = 2cos 450 = 2 x =Sin 450 = =y = 2sin 450 = 2 x =Translation factor (, )(a). = +=: . The origin is translated to ( , )(b). = + =: . (2, –1) is translated to ( (+ 2), (- 1))A translation takes the point(3, 2) to (-4, -5), Find where it takes(0 , 0)Solution= + where is translation factorENLARGEMENTEnlargement is a transformation in which a figure is made larger (magnified) or made smaller (diminished).– The number that magnifies or diminishes a figure is called the enlargement factor usually denoted by letter K. If K is less than 1 the figure is diminished and if it is greater than 1 the figure is enlarged K times.– In case of closed figures if the lengths are enlarged by a factor K then the area is enlarged by K2Examples: –Draw a triangle PQR with vertices P (0,0), Q (0, 3) and R (3, 0)P’ = 2 (0,0) = (0,0)Q’ = 2 (0,3) = (0,6)R’ = 2 (3,0) = (6,0)From the above question, what is the area of the new (enlarged) triangle?Solution.Area of the original triangle= x 3 x 3 = 4.5 square unitsThe area of the new triangle = 4.5 x K2= 4.5 x 22= 18 square unitsThe line segment AB with coordinated A (4,0) and B (0,3) enlarge to A΄B΄ by a factor 2. Find the coordinates for A΄ and B΄A’ = 2 (4, 0) = (8,0) B’=2(0,3) =(0,6)Find the image of the circle of radius one unit having its centre at (1,1) under enlargement transformation factor 5Solution:= 5(1,1)= (5,5)The image of the enlarged circle is (5,5)EXERCISE 2(a) Δ ADE to Δ ABC?(b) Δ ADE to Δ AFG?(a) Δ ADE to Δ ABC ==2.The point P(6,2) is enlarged by factor of 4, what is the new end point?Solution4 (6,2)= (24, 8):. The point is (24, 8)ABCD is a parallelogramSolutionEXERCISE 3List 3 examples of isometric transformationo Translation o Rotation o ReflectionIs enlargement an Isometric transformation? Enlargement is not an Isometric transformation.3.Find the image of the point Q (6, –8) after a rotation of 900 about theR90º(6, –8) = (–6,8)Draw a parallelogram ABCD with vertices A (2,5), B (5,5) , C (6,8), D (3,8) find and draw the image parallelogram formed by the translation wich moves the origin to (2,4)SolutionA = = +A =A = (4,9)B = = +=B = (7, 9)C = = +=C = (8, 12)D = + =D = (5, 12) ALL NOTES FOR ALL SUBJECTS QUICK LINKS:AGRICULTURE O LEVEL PURE MATHEMATICS A LEVELBAM NOTES A LEVELBASIC MATH O LEVELBIOLOGY O/A LEVELBOOK KEEPING O LEVELCHEMISTRY O/A LEVELCIVICS O LEVELCOMPUTER(ICT) O/A LEVELECONOMICS A LEVELENGLISH O/A LEVELCOMMERCE O/A LEVELACCOUNTING A LEVELGENERAL STUDIES NOTESGEOGRAPGY O/A LEVELHISTORY O/A LEVELKISWAHILI O/A LEVELPHYSICS O/A LEVELMOCK EXAMINATION PAPERSNECTA PAST PAPERS Basic Mathematics Study Notes Form 2 Basic Mathematics Study Notes Msomi Maktaba All Notes FORM 2MATHEMATICSPost navigationPrevious postNext postRelated Posts Chemistry Study Notes Form 3 Chemistry – QUANTITATIVE ANALYSIS AND VOLUMETRIC ANALYSIS November 12, 2018February 13, 2019ALL NOTES FOR ALL SUBJECTS QUICK LINKS: AGRICULTURE O LEVEL PURE MATHEMATICS A LEVEL BAM NOTES A LEVEL BASIC MATH O LEVEL BIOLOGY O/A LEVEL BOOK KEEPING O LEVEL CHEMISTRY O/A LEVEL CIVICS O LEVEL COMPUTER(ICT) O/A LEVEL ECONOMICS A LEVEL ENGLISH O/A LEVEL COMMERCE O/A LEVEL ACCOUNTING A LEVEL… Read More Form 6 Geography Study Notes Advanced Geography 2 (Geography Form 6) – AGRICULTURAL DEVELOPMENT February 25, 2019August 8, 2019ALL NOTES FOR ALL SUBJECTS QUICK LINKS: AGRICULTURE O LEVEL PURE MATHEMATICS A LEVEL BAM NOTES A LEVEL BASIC MATH O LEVEL BIOLOGY O/A LEVEL BOOK KEEPING O LEVEL CHEMISTRY O/A LEVEL CIVICS O LEVEL COMPUTER(ICT) O/A LEVEL ECONOMICS A LEVEL ENGLISH O/A LEVEL COMMERCE O/A LEVEL ACCOUNTING A LEVEL… Read More English Language Study Notes Form 4 English – READING LITERARY WORKS November 13, 2018February 13, 2019INTRODUCTION TO LITERATURE Literature – is the artistic expression of people‘s ideas which reflect social realities done through the use of language either in written or spoken. 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