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Advanced Mathematical Transformations Project: Vertex Matrix and Campus Analysis

This project focuses on using matrix transformations to analyze and represent a campus layout. It includes tasks such as moving, dilating, reflecting, and rotating the campus represented in a polygon form using vertex matrices. Various mathematical operations such as determining new coordinates after transformations and calculating areas using determinants are performed. Additionally, it addresses practical applications, such as the cost of items from a campus canteen, calculated through a system of equations. This comprehensive project showcases advanced mathematical concepts applied in real-world scenarios.

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Advanced Mathematical Transformations Project: Vertex Matrix and Campus Analysis

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  1. Done By: Advanced Math Project Essa Essa AhMEDAT1000198 Abdulla Nasser AT1000287 Ahmed ObaidAT1000272 AmerAbdulsalam AT1000268 Supervised By : Mister. YahyaAlsakaji Term 1 2011-2012

  2. Task1 (-8,9) (9 , 9) (-8,-7) (9,-7)

  3. Task2 Represent the campus (polygon) by vertex matrix. A (2,16) B(19,3) C(26,27) D(28,21) [ ] 2 19 26 28 16 3 27 21 Task 3 Use the matrix to perform the following transformations. [ • a. The campus is moved by 2 units right and 3 units down. Find the coordinates of the new location. ] [ ] [ ] 4 21 28 30 2 19 26 28 2 2 2 2 = + -3 -3 -3 -3 13 0 24 18 16 3 27 21 A’ (11,6) B’(-6,6) C’(-6,-10) D’(11,-10)

  4. Task3 Use the matrix to perform the following transformations. • b. Dilate the campus image so the perimeter will become 3 times more previous perimeter. What are the coordinate of the vertices of the new image? [ ] ] [ 6 57 78 84 2 19 26 28 3 = 48 9 81 63 16 3 27 21 A’ (27,27) B’(-24,27) C’(-24,-21) D’(27,-21) • c. Find the coordinates of the vertices of the original image of the campus after reflection in y=x ] [ ] [ ] [ 2 19 26 28 16 3 27 21 0 1 = 16 3 27 21 2 19 26 28 1 0 A’ (16,2) B’(3,19) C’(27,26) D’(21,28)

  5. Task 3 Use the matrix to perform the following transformations. • d. Find the coordinates of the vertices of the original image of the campus after it is rotated 270° counterclockwise about the origin. ] [ ] [ ] [ 2 19 26 28 16 3 27 21 0 1 = 16 3 27 21 -2 -19 -26 -28 -1 0 • e. Choose three vertices on the campus image that contains a land mark on the campus, use the determinant to find its area. The three vertices : (5,21) , (9,18) , (10,21) 5 21 1 area 9 18 1 1/2 = 7.5 units2 = 1/2 = (15) 10 21 1

  6. Task 4 Your campus canteen sells bottle of orange juice and apples. Ahmed bought 2 apples and 1 bottle of orange juice for 10 DHs. Abdullah bought 3 apples and 2 bottles of orange juice for 17 DHs. What is the cost of 1 apple and 1 bottle of orange juice? 2x + y = 10 3x + 2y = 17 Let apple be x Let bottle of orange juice be y 2 1 10 1 2 10 |c| = |x| = |y| = 3 2 17 2 3 17 = = = 2(2)-3(1) 10(2)-17(1) 2(17)-3(10) = = 4 3 = 1 = x |X|/|C| = 3/1 = 3 y = |y|/|C| = 4/1 = 4 1 apple costs 3DHS , and 1 bottle of orange juice costs 4DHS

  7. Task 5 IAT canteen Menu Make a 3 by 2 matrix to organize these data, and then use that matrix to find the prices after discount of 20%. [ ] 5 3 4 2 5 3 [ ] [ ] [ ] 5 3 5 3 4 2.4 4 2 - 0.20 4 2 = 3.2 1.6 5 3 5 3 4 2.4 [ ] [ ] 5 3 4 2.4 0.80 4 2 = 3.2 1.6 5 3 4 2.4

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