Theory question 114
Vectors and Transformations(Questions 14 and 15 are for candidates in Ghana only.)
(a) Using a scale of 2 cm to 4 units on both axes, draw on a graph sheet, two perpendicular axes Ox and Oy for the intervals −16 ≤ x ≤ 16, −16 ≤ y ≤ 16.
(b) Draw the triangle ABC with the vertices A(2, 2), B(8, 2) and C(6, 6).
(c) Draw the image ΔA1B1C1 of ΔABC through a rotation of 180° about the origin where A → A1, B → B1 and C → C1.
(d) Draw the image ΔA2B2C2 of ΔABC under the translation by the vector (−8, 8) where A → A2, B → B2 and C → C2.
(e) Draw the image ΔA3B3C3 of the ΔA1B1C1 under the reflection in the line x − 2 = 0, where A1 → A3, B1 → B3 and C1 → C3.
(f) Describe the transformation that will map ΔABC onto ΔA3B3C3.
Show the model answer
(a), (b) Axes to scale; A(2, 2), B(8, 2), C(6, 6).
(c) (x, y) → (−x, −y): A1(−2, −2), B1(−8, −2), C1(−6, −6).
(d) Add (−8, 8): A2(−6, 10), B2(0, 10), C2(−2, 14).
(e) Reflection in x = 2: (x, y) → (4 − x, y): A3(6, −2), B3(12, −2), C3(10, −6).
(f) ABC → A3B3C3 is (x, y) → (x + 4, −y): a reflection in the x-axis followed by a translation of 4 units in the positive x-direction (vector (4, 0)).
Write each transformation as a rule on (x, y) and combine them.