J20
(a) 3 0 0 1 0 2 0 0 1 0 0 3 3 0 = . Therefore, the image has vertices at: 1 1 0 0 2 2 O(0, 0), I1(3, 0), K1(3, 2) and J1(0, 2). (b) The determinant of N is (3 × 2) – (0 × 0) = 6. (c) The area of OI1K1J1 = 3 × 2 = 6 square units. The area of the unit square is 1 square unit. The area of the image is 6 square units. From this information we have the area of the image the areaof the object is the same as the determinant of the transformation matrix. Exercise 1. Find the area of the image of the unit square under the transformations represented by each of the following matrices. (a) 2 0 2 6 (b) 0 2 1 5 (c) 5 1 1 3 (d) 0 4 0 1 (e) 3 0 1 0 (f) 0 1 0 1 (g) 4 6 1 3 2 (h) 3 2 . 6 4 2. A rectangle with vertices at A(0, 0), B(4, 0), C(4, 3) and D(0, 3) is 1 2 transformed under the matrix . 1 4 (a) Find the coordinates of the vertices of the image, A1B1C1D1. (b) Draw rectangle ABCD and its image on the same axes.
(c) Find the areas of rectangles ABCD and A1B1C1D1. 3. A rectangle has vertices (1, 1), 1, 3), (4, 1) and (4, 3). Find the area of its image under the transformation represented by each of the following matrices. (a) 1 0 2 1 (b) 0 4 3 3 (c) 4 0 1 0 1 (d) 2 8 0 1 3 0 4. Under a transformation represented by matrix N = , rectangle A1B1C1D1 with 0 2 vertices at A1(6, 2), B1(6, 6), C1(15, 6) and D1(15, 2), is the image of rectangle ABCD. (a) Determine the area of rectangle ABCD. (b) Find the coordinates of A, B, C and D. 1 2 5. The matrix represents a transformation R. 0 1 (a) Find the images of the following points under R. (i) (1, 2) (ii) (-1, -1). (b) Describe transformation R fully. 6. Points A(-5, 1), B(-1, -1) and C(2, 5) are three vertices of rectangle ABCD. (a) Determine the coordinates of D. (b) Describe fully a single transformation (not a reflection) that maps A onto D, and B onto C. 7. Triangle T has vertices (2, 1), (4, 1) and (3, 3). (a) Find the area of the triangle. (b) Find the area of the image of T when it is transformed under the matrix: 0 1 2 2 (i) (ii) 1 0 1 3 3 4 6 0 (iii) (iv) 1 3 1 . 4 3 8. Triangle K, whose vertices are P(2, 3), Q(5, 3) and R(4, 1), is mapped onto triangle K1 whose vertices are P1(-4, 3), Q1(-1, 3) and R1(x, y) by a transformation given by a b matrix M = . c d (a) Find the: (i) elements of matrix M (ii) coordinates of R1. (b) Triangle K2 is the image of triangle K1 under a reflection in the line y = x. Find a single matrix that maps K onto K2.
- 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: Solution (a) The matrix that maps
- Page 25 and 26: (i) Find the coordinates of A 4, B
- Page 27: 25. (a) Draw the x and y axes from
(c)<br />
Find the areas of rectangles ABCD and A1B1C1D1.<br />
3. A rectangle has vertices (1, 1), 1, 3), (4, 1) and (4, 3). Find the area of its image<br />
under the transformation represented by each of the following matrices.<br />
(a)<br />
1<br />
0<br />
2<br />
1<br />
<br />
(b) <br />
0<br />
4<br />
3<br />
3<br />
(c)<br />
4<br />
0 <br />
<br />
1<br />
0<br />
1<br />
(d) <br />
<br />
2<br />
8 <br />
0 1<br />
3<br />
0 <br />
4. Under a transformation represented by matrix N = , rectangle A1B1C1D1 with<br />
0<br />
2<br />
vertices at A1(6, 2), B1(6, 6), C1(15, 6) and D1(15, 2), is the image of rectangle<br />
ABCD.<br />
(a) Determine the area of rectangle ABCD.<br />
(b) Find the coordinates of A, B, C and D.<br />
1<br />
2<br />
5. The matrix represents a transformation R.<br />
0<br />
1<br />
(a) Find the images of the following points under R.<br />
(i) (1, 2) (ii) (-1, -1).<br />
(b) Describe transformation R fully.<br />
6. Points A(-5, 1), B(-1, -1) and C(2, 5) are three vertices of rectangle ABCD.<br />
(a) Determine the coordinates of D.<br />
(b) Describe fully a single transformation (not a reflection) that maps A onto D,<br />
and B onto C.<br />
7. Triangle T has vertices (2, 1), (4, 1) and (3, 3).<br />
(a) Find the area of the triangle.<br />
(b) Find the area of the image of T when it is transformed under the matrix:<br />
0<br />
1<br />
<br />
2<br />
2<br />
(i) <br />
(ii) <br />
1<br />
0<br />
1<br />
3 <br />
3<br />
4<br />
6<br />
0 <br />
(iii) <br />
(iv) <br />
1<br />
3<br />
1 .<br />
<br />
4<br />
3 <br />
8. Triangle K, whose vertices are P(2, 3), Q(5, 3) and R(4, 1), is mapped onto triangle<br />
K1 whose vertices are P1(-4, 3), Q1(-1, 3) and R1(x, y) by a transformation given by<br />
a<br />
b <br />
matrix M = .<br />
c<br />
d <br />
(a) Find the:<br />
(i) elements of matrix M<br />
(ii) coordinates of R1.<br />
(b) Triangle K2 is the image of triangle K1 under a reflection in the line y = x.<br />
Find a single matrix that maps K onto K2.
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