☰ Contents · Geometry

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
13

Thales’ theorem

Textbook: pp. 40–42
GoalKnow Thales’ theorem and use it to divide a segment into equal parts.
New words
Thales’ theorem · Fales teoremasiparallel lines · parallel to‘g‘ri chiziqlartransversal · kesuvchiequal segments · teng kesmalar
Explanation

First the simple case: if parallel lines l1, l2, l3 cut equal segments on a line a, and b is parallel to a, they cut equal segments on b too (opposite sides of a parallelogram are equal). Thales’ theorem is more general: if parallel lines cutting the sides of an angle cut equal segments on one side, they cut equal segments on the other side too. The theorem generalizes the midline theorems for the triangle and the trapezoid. In the condition, the angle may be replaced by any two lines. The theorem is used to divide a segment into any number of equal parts: lay n equal segments along a ray, join the last point to the end of the segment and draw parallels to that line through the other points.

Worked examples
Equal segments of 4 cm are laid along one side of an angle and parallel lines are drawn through their ends. If the first segment on the other side is 5 cm, the second and third are also 5 cm, so the three make 15 cm.
To divide AB into 3 equal parts: draw a ray from A at an angle to AB, lay 3 equal segments on it, join the last point to B and draw lines parallel to this line through the other two points. They divide AB into 3 equal parts.
Class activity

“Divide the segment”: use the lines of a notebook (they are parallel) to divide a given segment into 5 equal parts.

Practice
1
Parallel lines cut equal segments of 6 cm on one side of an angle. On the other side the first segment is 9 cm. How long is the second segment (cm)?
2
Parallel lines cut 4 equal segments on one side of an angle. On the other side the first 3 of them total 21 cm. How many cm do all 4 total?
3
How many equal segments are laid along the ray to divide a segment into 5 equal parts?
4
Why does Thales’ theorem let us divide a segment into equal parts?