☰ Contents · Geometry

Motion, translation and symmetry about a line

Lessons 15–16 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
15

Geometric transformations of the plane. Motion and translation

Textbook: pp. 48–49
GoalKnow the concepts of geometric transformation, motion and translation; apply the translation formulas.
New words
geometric transformation · geometrik almashtirishmotion (isometry) · harakattranslation · parallel ko‘chirishvector · vektor
Explanation

Moving every point of a figure by some rule to get a new figure is a geometric transformation. A transformation that preserves distances is called a motion: XY = X₁Y₁. Under a motion a line goes to a line, a segment to an equal segment, an angle to an equal angle, a triangle to a congruent triangle; two motions in a row give again a motion. Figures that can be carried onto each other by a motion are called equal (congruent). In a translation along vector AB each point X goes to X₁ with XX₁ = AB. If the vector is (a; b), the point X(x; y) goes to X₁(x + a; y + b): x₁ = x + a, y₁ = y + b.

Worked examples
Along the vector (4; −1) the point P(2; 3) goes to x₁ = 2 + 4 = 6, y₁ = 3 + (−1) = 2, that is P₁(6; 2).
If a translation takes A(1; 1) to B(4; −3), the vector is (4 − 1; −3 − 1) = (3; −4).
Class activity

“The mover”: draw a triangle on a coordinate plane, translate it along (3; −2) and measure the sides of both triangles to check that they are equal.

Practice
1
Where does A(1; 2) go under translation along (−3; 5)? Give the new x-coordinate.
2
Under translation along (2; −5), where does the origin go?
3
A translation takes M(7; 1) to N(3; 4). What is the vector?
4
Why is a translation a motion?