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Problem solving
Textbook: p. 104
GoalKnow how to solve practical and geometric problems that use the Pythagorean theorem.
New words
practical problem · amaliy masalaequilateral triangle · teng tomonli uchburchakisosceles trapezoid · teng yonli trapetsiyasetting upright · tik o‘rnatish
Explanation
The Pythagorean theorem is useful in practical work too. For example, the triple 3, 4, 5 is used to check a post or the corner of a wall: measure 3 equal units along one line and 4 along the other; if the marks are 5 units apart, the angle is right. Whether something stands upright should be checked in more than one direction. In an equilateral triangle the altitude halves the base, so h² = a² − (a : 2)². Al-Khwarizmi also found the area of an equilateral triangle by getting its altitude this way. In an isosceles trapezoid a leg, the altitude and half the difference of the bases form a right triangle.
Worked examples
An equilateral triangle has side 8: the altitude squared is 64 − 16 = 48.
An isosceles trapezoid has bases 10 and 20 and legs 13. Half the difference of the bases is 5, the altitude is 12 and the area is 15 · 12 = 180.
Class activity
“Upright post”: with an adult, stand a long stick upright in the garden and check it with the 3-4-5 method in two directions.
Practice
1
An isosceles trapezoid has bases 8 and 18 and legs 13, and its altitude is 12. What is its area?
156
2
What is the square of the altitude of an equilateral triangle with side 14?
147
3
A right trapezoid has bases 9 and 18, the longer leg 15 and altitude 12. What is its area?
162
4
Why must the upright position of a post be checked in at least two directions?
Even if it is straight in one direction, the post may still lean in another direction.