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Thales’ theorem and its use

Lessons 13–14 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
14

Problems using Thales’ theorem

Textbook: pp. 43–47
GoalUse ratios of segments, proportional segments and the bisector property.
New words
ratio of segments · kesmalar nisbatiproportional segments · proporsional kesmalarproportion · proporsiyabisector · bissektrisa
Explanation

If we measure two segments in the same unit and divide one number by the other, we get their ratio; it tells how many times one segment is longer or shorter than the other. If we change the unit (say, mm instead of cm), both numbers change by the same factor, so the ratio stays the same. If AB : BC = A1B1 : B1C1, the segments AB, BC are called proportional to A1B1, B1C1. Theorem: if two parallel lines cross both sides of an angle, the segments they make on the sides are proportional; with vertex O, OA : OB = OC : OD. In a triangle, an angle bisector cuts the opposite side into two parts whose ratio equals the ratio of the other two sides.

Worked examples
Parallel lines cut the sides of an angle so that OA : AB = 3 : 4. If OC = 12 cm on the other side, OC : CD = 3 : 4 gives CD = 16 cm.
In triangle ABC, AB = 12 cm, AC = 18 cm and BC = 20 cm. The bisector of angle A divides BC with BD : DC = 12 : 18 = 2 : 3, so BD = 8 cm and DC = 12 cm.
Class activity

“Proportion cards”: teams arrange cards with the parts of a proportion in the right order to form a proportion of segments.

Practice
1
Segments a = 4 cm, b = 6 cm and c = 10 cm are given. How long is d (cm) if a : b = c : d?
2
Parallel lines cut the sides of an angle: OA = 6 cm, OB = 15 cm, OC = 8 cm. How long is OD (cm), given OA : OB = OC : OD?
3
In a triangle AB = 10 cm, AC = 15 cm, BC = 20 cm. How long is the part BD cut off BC by the bisector of angle A (cm)?
4
Why does the ratio of segments not depend on the unit?