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19. Using a scale of 1 cm to one unit in each case draw the axes taking values of x from -4 to +6 and values of y from 0 to 12. (a) Draw and label the quadrilateral OABC with O(0, 0), A(2, 0), B(4, 2), C(0, 2) (b) Find and draw the image of OABC under the transformation whose matrix is 2.4 1.8 R, where R = . 1.8 2.4 (c) Calculate, in surd form, the lengths OB and O ′ B′ (d) Calculate the angle AO A′ . (e) Given that the transformation R consists of a rotation about O followed by an enlargement, state the angle of the rotation and the scale factor of the enlargement. cos sin 20. The matrix R = represents a positive rotation of θ 0 about sin cos the origin. Find the matrix which represents a rotation of: (a) 90 0 (b) 180 0 (c) 30 0 (d) -90 0 (e) 60 0 (f) 150 0 (g) 45 0 (h) 53.1 0 Confirm your results for parts (a), (e), (h) by applying the matrix to the quadrilateral O90, 0), A(0, 2), B(4, 2), C(4, 0). x ′, y′ of a point (x, y) under a transformation is given by x 3 0 x 2 = + y 1 2 y 5 (a) Find the coordinates of the image of the point (4, 3). (b) The image of the point (m, n) is the point (11, 7). Write down two equations involving m and n and hence find the values of m and n. (c) The image of the point (h, k) is the point (5, 10). Find the values of h and k. 21. The image ( ) 22. Draw A(0, 2), B(2, 2), C(0, 4) and its image under an enlargement, A′(2, 2), B′(6, 2), C′ (2, 6). (a) What is the centre of enlargement? (b) Find the image of ABC under a reflection in the line x = 0 (c) Find the translation which maps this image onto A′ B′ C′ . (d) What is the matrix X and vector v which represents a reflection in the line x = 2? 2 0 1 0 23. A = ; h and k are numbers so that A 2 = hA + kI, where I = . 1 2 0 1 Find the values of h and k. a 1 1 0 24. M = . Find the values of a if: (a) M 2 = 17 1 a 0 1 (b) M = -10.
25. (a) Draw the x and y axes from -5 to 5 using a scale of 1 cm to represent I unit on each axis. Draw triangle ABC with A(1, 1), B(4, 1) and C(4, 2). (b) (i) Draw the image of triangle ABC when it is rotated 90 0 anticlockwise about the origin. Label this image A1B1C1. (ii) 3 Triangle A1B1C1 is translated by the vector . Draw and label this 1 image A2B2C2. (iii) Describe fully a single transformation which maps triangle ABC onto A2B2C2. (c) (i) Draw the image of triangle ABC under the transformation represented 1 0 by the matrix . Label this image A3B3C3. 0 2 (ii) Describe fully the single transformation which maps triangle ABC onto triangle A3B3C3.
Page 1 and 2: Matrices and Transformations There
Page 3 and 4: Reflections Under a reflection: (a)
Page 5 and 6: Shears A shear is a transformation
Page 7 and 8: Consider a reflection of the unit s
Page 9 and 10: a b The matrix transforms the vec
Page 11 and 12: 2a + 3b = 8 ………(ii) 3a + b =
Page 13 and 14: (a) Triangle ABC with vertices A(2,
Page 15 and 16: Example Triangle B has vertices at
Page 17 and 18: under the following combinations of
Page 19 and 20: Alternatively, this transformation
Page 21 and 22: Solution (a) The matrix that maps
Page 23 and 24: (c) Find the areas of rectangles AB
Page 25: (i) Find the coordinates of A 4, B

J20

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