Lessons 18–19 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
18
Transformations of space and similarity
Textbook, Part 1: pp. 133–141
GoalKnow the transformations of space (translation, central, plane and axial symmetry, rotation, homothety) with their coordinate formulas, and distinguish similarity from motion.
A motion (isometry) is a transformation that preserves distances between points; translation, symmetry in a point, a plane or a line, and rotation are such transformations. Under translation by the vector p(a, b, c) the point (x, y, z) goes to (x + a, y + b, z + c). Under central symmetry about the point C the point A goes to A′ with C the midpoint of AA′: A′ = 2C − A. For symmetry in the coordinate planes one coordinate changes sign: in Oxy it is (x, y, −z), in Oxz (x, −y, z), in Oyz (−x, y, z); for symmetry in a coordinate axis two coordinates change sign (about Oz: (−x, −y, z)), and for symmetry in the origin all three do. A homothety with centre O and ratio k ≠ 0 sends X to X′ with OX′ = k·OX; distances are multiplied by |k| and the shape stays similar (a homothety is a similarity transformation, and is not a motion unless k = ±1). Two figures are similar if one is mapped onto the other by a similarity transformation.
Worked examples
Translation by p(2, −1, 4): (3, 0, −2) → (5, −1, 2). The point symmetric to A(4, 0, 5) about C(1, 2, 3) is A′ = 2C − A = (2 − 4, 4 − 0, 6 − 5) = (−2, 4, 1). Check: the midpoint of AA′ is ((4 − 2)/2, (0 + 4)/2, (5 + 1)/2) = (1, 2, 3) = C.
A homothety with centre at the origin and k = −2 sends (1, −3, 2) → (−2, 6, −4). The distance between (0, 0, 0) and (1, 0, 0) is 1; between their images (0, 0, 0) and (−2, 0, 0) it is 2: the distance is multiplied by |k| = 2, so a homothety is not a motion. Under symmetry in Oyz, (4, −1, 5) → (−4, −1, 5); about the Oz axis, (4, −1, 5) → (−4, 1, 5).
Class activity
“Mirror and copy”: students make a paper cube model (cut with scissors only with an adult’s help) and place one vertex at the origin. Each pair picks three transformations (translation, plane symmetry, homothety), computes the image vertices’ coordinates and checks them with another pair.
Practice
1
Under translation by p(−3, 2, 5), where does (1, 1, 1) go?
(−2, 3, 6)
2
Find the point symmetric to (2, −5, 4) in the plane Oxz.
(2, 5, 4)
3
Find the point symmetric to A(5, −1, 2) in the centre C(1, 3, −4).
A′ = 2C − A = (−3, 7, −10)
4
Why is a homothety with k = 3 not a motion?
A motion preserves distances, while this homothety triples them: (0, 0, 0), (1, 0, 0) → (0, 0, 0), (3, 0, 0), so the distance changes from 1 to 3.