Project work

A PROJECT REPORT on Finding the image of a given object after reflection, rotation, translation, and enlargement(reduction) Submitted to: Yam Pandeya Sindhuli Vidhyashram Secondary School Kamalamai – 4, Dovantar, Sindhuli Submitted by: Paras Ray Class: IX Roll No: 12 In Partial Fulfillment of Second Terminal Examination of Optional Mathematics Phalgun, 2077

Acknowledgment At first, I would like to express my special thanks of gratitude to my subject teacher Mr. Yam Pandeya who gave me the golden opportunity to do this wonderful project work on the topic \"Transformation\" which taught me so many new things. Secondly, I would also like to thank my school principal Mr. Posak Poudel and all the teaching and non-teach staff of this school who helped me a lot in finalizing this project within the limited time frame, despite their busy schedules. Similarly, I would like to thank my friends who helped me a lot in gathering different information, collecting data, and guiding me from time to time in making this project they gave me different ideas in making this project unique. Thanking you, Pares Ray Class- IX EVALUATION AND APPROVAL This project report has been evaluated and approved. …….……………… ………….……………… Mr. Mr Yam Pandeya Mr. Posak Paudel Subject Teacher Principal i

Table of content Acknowledgment i Evaluation and approval ii Introduction 1 1 Objectives of project 1-6 1 Isometric transformation 2 3 Reflection 3 Properties of reflection 4 4 Reflection using coordinate 5 5 Translation 6 6-8 Properties of translation 7 7 Translation using coordinates 8 8 Rotation Properties of rotation Rotation using coordinate Non-Isometric transformation Enlargement Properties of enlargement Enlargement using coordinates Combination of transformation ii

Introduction Transformation involves moving an object from its original position to a new position. A transformation transforms or changes the position, shape, or size of a geometric figure. A figure before the transformation is called an object and After the transformation is called an image. Each point in the object is mapped to a corresponding point in the image. Objective of Project The main objectives of this projects were as follows: i) Identification of Isometric and non-Isometric transformation using our daily life. ii) To find the image of a given object after reflection, rotation, translation, and enlargement(reduction) iii) Uses of the appropriate criterion of transformation for the given daily life problem. Types of transformations: In these lessons, we will study the following types of transformations in math: i) Isometric Transformation ii) Non-Isometric Transformation Isometric Transformation The transformation in which the size of the figure remains the same and only the position of the figure is changed after transformation is called isometric transformation. In isometric transformation the object and its image are congruent. Translation, reflection, and rotations are called isometric transformations because the image is the same size and shape as the original object. The original object and the image are congruent. Reflection involves “flipping” the object over a line called the line of reflection. The reflection has a mirror image about a line as a two-way mirror. A line that plays the role of a two-way mirror to a given image of the given object is called the axis of reflection. 1

Properties of Reflection: a. The object and the image under the reflection are congruent. b. The object and its image will be equal to the axis of reflection. c. The line joining the object and image will be perpendicular to the axis of reflection. 2

Reflection Using Co-ordinates P'(x, -y) P'(-x, y) Reflection on x-axis P'(y, x) P (x, y) Reflection on y-axis P (x, y) Reflection on line y = x P (x, y) Reflection on line y= - x P'(-y, -x) P (x, y) P (x, y) Reflection on line x = - h P'(2h - x, y) P (x, y) Reflection on line y = - k P'(x, 2k -y) Translation involves “sliding” the object from one position to another. In translation, each point of the given object is displaced through definite distance and direction. The displacement is defined by a vector (������������). 3

Properties of Translation: a. The object and the image under the translation are congruent. b. The lines joining any point of the object with its corresponding image are parallel and equal. Translation Using Co-ordinates P (x, y) T = (������������) P'(x + a, y + b) 4

Rotation involves “turning” the object about a point called the center of rotation. In rotation, each point rotates through an angle about a fixed point in the given direction. The fixed point is called the center of rotation and the angle is called the angle of rotation. Properties of Rotation a. The object and image under rotation are congruent. b. Every point on the object is rotated at trough same angle in the same direction about the Centre of rotation to reach its image. 5

Rotation Using Co-ordinates P (x, y) R [90o or -270o, C (0, 0)] P'( -y, x) P'(y, -x) R [-90o or 270o, C (0, 0)] P'(-x, y) P (x, y) P'(a + b -y, -a +b + x) P'(a - b + y, a +b - x) R [180o, C (0, 0)] P'(2a - x, 2b - y) P (x, y) R [90o or -270o, C (a, b)] P (x, y) R [-90o or 270o, C (a, b)] P (x, y) R [180o, C (a, b)] P (x, y) Non-Isometric Transformation The transformation in which the size of the figure after transformation is changed is called non-isometric transformation. Enlargement and the non-isometric transformation. The object and image are in this transformation. Enlargement (Reduction) involves a resizing of the object. It could increase in size (enlargement) or a decrease in size. In this transformation, the size of an object is changed without changing its shape if the size of the object increase, we call it an enlargement and if the size of an object decreases, we call it a reduction. The enlargement is made with the help of a fixed point is called the Centre of enlargement and the ratio of the corresponding sides of the image and object is called the scale factor. 6

Properties of enlargement a. The object and image under enlargement are similar. Length of the side of image figure b. Scale factor = Length of the corresponding side of object figure c. If the scale factor k > 1, then the transformation is called enlargement. d. If scale factor 0 < k < 1, then the transformation is called reduction. e. If k = 1, then the transformation is identity. f. If the scale factor k < 0, then the image will be on the opposite side of the object from the Centre of enlargement. 7

Enlargement or Reduction Using Co-ordinates E [(0, 0), k)] P (x, y) P'(kx, ky) E [(a, b), k] P (x, y) P'[k(x -a) + , (y - k) + b] Combination of Transformation The transformations discussed above are examples of single transformation. When an object has been transformed, its image can again be transformed to form a new image. Such transformation is called a combination of transformations. After a combination of transformations, the final image can be described by a single transformation. The triangle ABC first reflected about line y = -x then translated by vector ������ = (−52) to obtain final image A\"B\" C\". 8

The quadrilateral ABCD is first reflected about the x-axis to obtain the quadrilateral A'B'C'D' and quadrilateral A'B'C'D' again rotated to obtain quadrilateral A\"B\" C\" D\". 9