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(c) Describe fully the single transformation that maps PQRS onto P2Q2R2S2. (d) Find the matrix of transformation that maps PQRS onto P2Q2R2S2. 3. Draw triangle T whose vertices are (4, 3), (-3, -1) and (-1, 3). (a) Triangle R is the image of triangle T under the transformation represented 0.6 0.8 by the matrix N = 0.8 0.6 (i) Calculate the coordinates of the vertices of triangle R. (ii) Draw and label triangle R (b) Describe fully a single transformation that maps triangle T onto triangle R. 4. Triangle XYZ is such that X(1, 4), Y(1, 7) and Z(3, 1) are its vertices. Triangle XYZ is mapped onto triangle X 1Y 1Z 1 by the transformation 0 1 represented by the matrix M = . 1 0 (a) Draw and label triangle X 1Y 1Z 1 and describe fully the transformation that maps triangle XYZ onto X 1Y 1Z 1 in geometrical terms. (b) Reflect ∆XYZ in the y-axis, and label the vertices of its image as X2, Y2, and Z2. Write down the elements of matrix N which represents this transformation. (c) Triangle X1Y1Z1 can be mapped onto triangle X2Y2Z2 by a single transformation P. (i) Describe this transformation fully. (ii) Write down the elements of matrix P which represents this transformation. (iii) State the relationship between M, N and P. The inverse of a transformation The inverse of a transformation reverses the transformation, i.e. it is the transformation which takes the image back to the object. For example, the inverse of an anticlockwise rotation of 90 0 about O is a clockwise rotation of 90 0 about O. 3 If translation T has vector , the translation which has the opposite effect has vector 2 3 . This is written as T -1 2 If M is a transformation representing a transformation, then the matrix representing the inverse of the transformation is indicated as M -1 . Example 1 2 Matrix N = describes a transformation on ∆PQR. The coordinates of the image 2 3 are P1(-3, 2), Q1(2, 1) and R1(0, 1). Find: (a) the matrix that maps ∆P1Q1R1 onto ∆PQR. (b) the coordinates of P, Q and R.
Solution (a) The matrix that maps ∆P1Q1R1 onto ∆PQR is the inverse of N. N -1 = 1 3 2 3 2 1 = 2 1 2 1 (b) Example P1 Q1 R1 P Q R 3 2 3 2 0 5 8 2 = 2 1 2 1 1 4 5 1 Therefore, the coordinates of PQR are P(-5, 4), Q(8, -5), R(2, -1) 2 1 A(2, 1), B(5, 1) and C(5, 3) are vertices of ∆ABC and M = . 3 2 (a) Find the image A1B1C1 of ABC under the transformation represented by matrix M. (b) Find M -1 . (c) Find the image of A1B1C1 under the transformation represented by matrix M -1 . (d) Comment on your results in (c). Solution A B C A1 B1 C1 2 1 2 5 5 5 11 13 (a) = Therefore, the coordinates of the 3 2 1 1 3 8 17 21 vertices of the image are: A1(5, 8), B1(11, 17) and C1(13, 21). Area and the determinant of a matrix. Under some transformations, the size and shape of objects do not change, while under other transformations they change. When a matrix represents a transformation, the ratio of the area of the image to the area of the object is equal to the determinant of the matrix of transformation. Example A unit square with vertices O(0, 0), I(1, 0), K(1, 1) and J(0, 1) is given a 3 0 transformation represented by matrix N = . 0 2 (a) Draw the square and its image. (b) Find the determinant of N. (c) Find the area of the image OI1K1J1. Solution O I K J O I1 K1 J1
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: Alternatively, this transformation
Page 23 and 24: (c) Find the areas of rectangles AB
Page 25 and 26: (i) Find the coordinates of A 4, B
Page 27: 25. (a) Draw the x and y axes from

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