Theory question 114
Vectors and Transformations(Questions 14 and 15 are for candidates in Ghana, Sierra Leone and The Gambia only.)
(a) Using a scale of 2cm to 1 unit on both axes, draw on a graph sheet, two perpendicular axes Ox and Oy, for the interval −5 ≤ x ≤ 5 and −6 ≤ y ≤ 6.
(b) Draw clearly, indicating the coordinates of all vertices (i) rectangle ABCD with vertices A(−2, −1), B(1, −1), C(1, 1) and D(−2, 1); (ii) the image A1B1C1D1 of ABCD under an enlargement from the origin O with scale factor 2 where A → A1, B → B1, C → C1 and D → D1; (iii) the image A2B2C2D2 of ABCD under a reflection in the line y = 2 where A → A2, B → B2, C → C2 and D → D2.
(c) Determine the equation of the line AC.
Show the model answer
(a), (b)(i) Axes to scale; rectangle ABCD plotted and labelled.
(ii) Multiply coordinates by 2: A1(−4, −2), B1(2, −2), C1(2, 2), D1(−4, 2).
(iii) Reflection in y = 2 maps (x, y) to (x, 4 − y): A2(−2, 5), B2(1, 5), C2(1, 3), D2(−2, 3).
(c) Gradient of AC = (1 − (−1))/(1 − (−2)) = 2/3. y + 1 = (2/3)(x + 2), i.e. 3y = 2x + 1.
A point and its image are the same distance from the mirror line y = 2.