56
Calculation problems
Textbook: pp. 146–147
GoalSolve calculation problems using a diagram and formulas.
New words
calculation problem · hisoblashga doir masalaratio · nisbatexterior angle · tashqi burchak30° angle property · 30° li burchak xossasi
Explanation
In a calculation problem the required quantity is found step by step from the given numbers and known formulas. First draw a diagram and introduce notation. If a ratio is given, write the parts as x, 2x and so on and form an equation. Frequently used facts: adjacent angles sum to 180°; the angles of a triangle sum to 180°; an exterior angle equals the sum of the two non-adjacent angles; in a right triangle the leg opposite a 30° angle is half the hypotenuse; a bisector halves an angle. At the end check that the answer fits the conditions.
Worked examples
Adjacent angles are in the ratio 4 : 5. 4x + 5x = 180°, 9x = 180°, x = 20°. The angles are 80° and 100°. Check: 80° + 100° = 180°.
A right triangle has an acute angle of 30° and the sum of the hypotenuse and the short leg is 27 cm. If the short leg is b, the hypotenuse is 2b: b + 2b = 27, b = 9, hypotenuse 18 cm.
Class activity
Sketch a simple diagram in your notebook, mark one unknown angle as x, and form and solve your own equation.
Practice
1
Adjacent angles are in the ratio 2 : 3. How many degrees is the smaller angle?
72
2
The apex angle of an isosceles triangle is 40°. How many degrees is a base angle?
70
3
Two angles of a triangle are 40° and 60°. What angle (in degrees) do their bisectors form where they meet?
130
4
Why may we write c = 2b for the leg opposite 30°?
Because in such a right triangle that leg is half the hypotenuse.