Theory question 115
Vectors and Transformations(a) Using a scale of 2 cm to 2 units on both axes, draw two perpendicular axes OX and OY for the interval −8 ≤ x ≤ 8 and −10 ≤ y ≤ 10.
(b) Draw, clearly labelling all the vertices and coordinates, (i) triangle PQR with P(−3, 6), Q(−1, 1) and R(4, 2); (ii) ΔP1Q1R1, with P1(5, 6), Q1(3, 1) and R1(−2, 2); (iii) ΔP2Q2R2, the image of ΔPQR under a rotation through 180° about the origin where P → P2, Q → Q2 and R → R2; (iv) the point Q3, the image of Q under a translation by the vector (−4, 3).
(c) What single transformation maps ΔPQR onto ΔP1Q1R1?
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(a), (b)(i), (ii) Axes to scale; plot and label PQR and P1Q1R1.
(iii) (x, y) → (−x, −y): P2(3, −6), Q2(1, −1), R2(−4, −2).
(iv) Q3 = (−1 − 4, 1 + 3) = (−5, 4).
(c) Each point keeps its y-coordinate and the midpoints of PP1, QQ1, RR1 all have x = 1: a reflection in the line x = 1.
Check the midpoint of each point and its image to find the mirror line.