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Test your knowledge
Textbook: pp. 71–73
GoalReview the rest of Chapter I (transformations, homothety, similarity) and prepare for test questions.
New words
motion · harakathomothety · gomotetiyasimilarity · o‘xshashlikscale · masshtab
Explanation
A quick recap of the chapter. A motion is a transformation that preserves distance: translation, reflection in a line, central symmetry and rotation. In a similarity transformation distances are multiplied by the same k; a homothety is a special case, where lengths scale by k and areas by k². In coordinates: about Ox (x; −y), about Oy (−x; y), about the origin (−x; −y), homothety (kx; ky), translation (x + a; y + b). In test questions first identify the transformation, then apply the matching formula. A common mistake is mixing up k and k².
Worked examples
Test: first translate (2; −5) along (3; 1): (5; −4). Then reflect about Ox: (5; 4).
Test: with homothety coefficient 2/3 the area is multiplied by 4/9: 45 cm² → 20 cm².
Class activity
“Test race”: 6 questions in 10 minutes; swap and check in pairs, noting the kind of error (k/k², sign, order).
Practice
1
Reflect (−4; 7) about the origin. What is the new y-coordinate?
-7
2
The homothety centre is the origin and k = 3. What is the x-coordinate of the image of (2; −1)?
6
3
Which of these are motions: homothety with k = 2, rotation, translation?
The rotation and the translation.
4
Why is a similarity with k ≠ 1 not a motion?
Distances are multiplied by k, not preserved.