Similarity of figures; properties of similar polygons
Lessons 18–19 · 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
18
Similarity of geometric figures
Textbook: pp. 58–59
GoalKnow the similarity of arbitrary figures (similarity transformation) and solve scale problems.
If every point of figure F goes to exactly one point of figure F* and the distance between any two points changes by the same factor k (X*Y* = k·XY, k > 0), the transformation is a similarity transformation and k is the similarity coefficient. Figures are similar if some similarity transformation carries one onto the other (F ∽ F*). Under a similarity a line goes to a line, a ray to a ray, an angle to an equal angle, and segment lengths are multiplied by k. If k = 1, the transformation is a motion. The scale of a map or plan is an example of a similarity coefficient: on a 1 : 20 000 map 1 cm stands for 20 000 cm = 200 m.
Worked examples
The map scale is 1 : 25 000. If two schools are 6 cm apart on the map, the real distance is 6 · 25 000 = 150 000 cm = 1.5 km.
A 2 × 5 cm rectangle becomes 6 × 15 cm with k = 3: the perimeter goes from 14 to 42 (3 times) and the area from 10 to 90 (9 times).
Class activity
“Floor plan”: measure the classroom with a tape and draw its plan on squared paper at 1 : 50; check by recalculating the real sizes from the plan.
Practice
1
The plan scale is 1 : 200. A room is 5 cm long on the plan. How long is it in reality (m)?
10
2
The scale is 1 : 1000. How many cm is a real distance of 350 m on the plan?
35
3
A triangle undergoes a similarity transformation with k = 3. How do its angles change?