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Projections of segments

Lessons 48 · 1 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
48

Projections of segments and proportionality

Textbook: pp. 130–131
GoalKnow the notion of the projection (shadow) of a segment on a line; use that projections of segments on parallel lines are proportional to the segments.
New words
projection of a segment · kesma proyeksiyasidropping a perpendicular · perpendikulyar tushirishproportional segments · proporsional kesmalarA₁B₁ = AB·cosα · A₁B₁ = AB·cosα
Explanation

Let a segment AB and a line m be given. If we drop perpendiculars AA₁ and BB₁ from A and B to m, the segment A₁B₁ is called the projection (shadow) of AB on m. If the angle between AB and m is α, the projection has length A₁B₁ = AB·cosα; if AB is parallel to m the projection equals AB, and if it is perpendicular the projection shrinks to a point. The projections on m of segments lying on one line or on parallel lines are proportional to the segments: A₁B₁ : AB = C₁D₁ : CD, because they make the same angle with m. So if the projection of one segment is known, we find the other by a proportion.

Worked examples
AB = 20 and CD = 30 lie on parallel lines, and the projection of AB is 12. The projection of CD is x: x/30 = 12/20, so x = 18.
The ends of AB are 5 and 11 from m (on the same side) and AB = 10: the difference is 6, so the projection is √(100 − 36) = 8.
Class activity

“Measure a shadow”: on a sunny day stand a stick upright and measure its shadow (never look straight at the sun!). Check that shadows are proportional to heights for a building and the stick.

Practice
1
A 14 cm segment makes a 60° angle with m. What is its projection?
2
Segments AB = 14 and CD = 21 lie on parallel lines; the projection of AB is 10. What is the projection of CD?
3
A 12 cm segment has a 6 cm projection on line m. What angle does the segment make with m?
4
Why is the projection of a segment perpendicular to m a point?