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 2 units on both axes, draw on a graph sheet two perpendicular axes OX and OY for the interval −8 ≤ x ≤ 8 and −8 ≤ y ≤ 8.
(b) Draw clearly and label the vertices as appropriate: (i) triangle PQR with P(1, 2), Q(5, 3) and RQ = (2, −3); (ii) the image ΔP′Q′R′ of ΔPQR under a rotation of 180° about the origin where P → P′, Q → Q′ and R → R′; (iii) the image ΔP″Q″R″ of ΔP′Q′R′ under a reflection in the line x = 0 where P′ → P″, Q′ → Q″ and R′ → R″.
(c) Describe fully a single transformation that maps ΔPQR onto ΔP″Q″R″.
Show the model answer
(b)(i) R = Q − RQ = (5 − 2, 3 + 3) = (3, 6). Plot P(1, 2), Q(5, 3), R(3, 6).
(ii) (x, y) → (−x, −y): P′(−1, −2), Q′(−5, −3), R′(−3, −6).
(iii) Reflection in x = 0 (the y-axis): (x, y) → (−x, y): P″(1, −2), Q″(5, −3), R″(3, −6).
(c) PQR → P″Q″R″ is (x, y) → (x, −y): a reflection in the x-axis (the line y = 0).
Combine the two rules: (x, y) → (−x, −y) → (x, −y).