Lessons 22 · 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
22
Practical exercise and application
Textbook: pp. 66–67
GoalApply similarity in practice: find a height with a stick or a shadow, find an inaccessible distance by building a similar triangle, and compute areas using a scale.
To find the height of a tall tower, a vertical stick is placed so that the shadow of its top and the shadow of the tower's top fall on the same point. The two right triangles formed are similar (common angle), so tower height H = stick height · (distance from tower base to shadow tip) : (distance from stick base to shadow tip). To find an inaccessible distance AB, choose a convenient point C and measure the segment AC and the angles A and C. Draw on paper a triangle A₁B₁C₁ with the same angles; by AA the triangles are similar, so AB = A₁B₁ · AC : A₁C₁. When an area is found from a scale picture, the ratio of lengths must be squared. Do measurements near rivers or roads only with adults and in a safe place.
Worked examples
Stick 1.5 m, 3 m from the stick base to the shadow tip, 240 m from the tower base to the shadow tip: H = 1.5 · 240 : 3 = 120 m.
If AC = 120 m, and on paper A₁C₁ = 60 mm and A₁B₁ = 78 mm, then AB = 78 · 120 : 60 = 156 m.
Class activity
“Schoolyard surveyors”: with adult permission and supervision in a safe place, find the distance to a faraway object using a similar triangle and compare it with a pace count.
Practice
1
A stick is 2 m tall, 5 m from its base to the shadow tip, and 40 m from the tree base to the shadow tip. How tall is the tree (m)?
16
2
AC = 90 m, A₁C₁ = 30 mm and A₁B₁ = 41 mm. How long is AB (m)?
123
3
On a map a lake is a 2 cm × 3 cm rectangle at scale 1 : 10 000. What is its real area (m²)?
60000
4
Why must the stick in the shadow method be vertical?
Otherwise the triangle is not right-angled and not similar to the tower's triangle.