Lessons 32–33 · 2 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
32
Solving triangles
Textbook: pp. 94–95
GoalSolving triangles: find the remaining sides and angles from the given elements with the laws of sines and cosines.
New words
solving a triangle · uchburchakni yechishelements of a triangle · uchburchak elementlariadjacent angles · yopishgan burchaklarthird side · uchinchi tomon
Explanation
The sides and angles of a triangle are its elements. Finding the remaining elements from the given ones is called solving the triangle. There are three main cases. 1) One side and two adjacent angles: complete the third angle to 180° and find the other sides with the law of sines. 2) Two sides and the included angle: find the third side by the law of cosines, then the other angles by the law of cosines or sines. 3) Three sides: find the angles with the law of cosines, starting with the largest angle since it may be obtuse. Values of non-special angles are found approximately with a calculator or a table.
Worked examples
a = 6√3, β = 30°, γ = 30°: α = 180° − 60° = 120°; b = a·sinβ/sinα = 6√3·(1/2)/(√3/2) = 6, c = 6.
a = 5, b = 8, γ = 60°: c² = 25 + 64 − 2·5·8·½ = 49, so c = 7. cosα = (64 + 49 − 25)/(2·8·7) = 11/14, α ≈ 38° and β = 180° − 60° − 38° ≈ 82°.
Class activity
“Triangle workshop”: each group picks one of the three cases, makes up its own problem and solves it; another group checks the solution by working backwards.
Practice
1
a = 12, β = 30°, γ = 60°. Find α and b.
α = 90°, b = 6
2
a = 3, b = 8, γ = 60°. What is c²?
49
3
What is the largest angle (in degrees) of a triangle with sides 7, 8, 13?
120
4
Why is it convenient to find the largest angle first when solving from three sides?
The sign of its cosine shows whether it is obtuse; the other two angles are then certainly acute.