The logic of planimetry and problem-solving methods
Lessons 16–17 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
17
Geometric problems and methods of solving them
Textbook, Part 1: pp. 102–107
GoalDistinguish the kinds of geometric problems and the methods of solving them (synthetic, analytic, contradiction, algebraic, area, vector, coordinate) and apply them to simple problems.
New words
synthetic method · sintetik usulanalytic method · analitik usulproof by contradiction · teskarisini faraz qilishcoordinate method · koordinatalar usuli
Explanation
By their demand geometric problems are to calculate, prove, investigate or construct. By structure the methods are synthetic (a chain of reasoning from the data towards the conclusion), analytic (starting from the conclusion and working backwards to what is needed) and proof by contradiction (assuming the conclusion fails and reaching a contradiction); the last rests on the law that of two contradicting statements exactly one is true. By mathematical tools there are the algebraic method (name the unknown with a letter and form an equation), the area method (write one figure’s area by two formulas and equate), the vector method, the coordinate method and the method of geometric transformations. The algebraic algorithm: draw a figure, name the unknown, form an equation, solve it, analyse the solution, write the answer. In the coordinate method we place the figure in the coordinate plane so that many points have zero coordinates.
Worked examples
Algebraic method: an isosceles triangle has perimeter 32 cm and its base is 4 cm shorter than a lateral side. Let the lateral side be x, the base x − 4: x + x + (x − 4) = 32 ⇒ 3x = 36 ⇒ x = 12. The sides are 12 cm, 12 cm, 8 cm. Analysis: 8 + 12 > 12, so the triangle exists.
Coordinate method: we prove that the diagonals of a rectangle are equal. Let A(0, 0), B(a, 0), C(a, b), D(0, b). AC = √(a² + b²), BD = √((0 − a)² + (b − 0)²) = √(a² + b²). Hence AC = BD.
Class activity
Each group gets the same problem (for example, the diagonals of a rectangle are equal) and solves it by a different method — synthetic, coordinate or vector — and then they discuss which was most convenient.
Practice
1
List the four kinds of geometric problems by their demand.
Problems to calculate, to prove, to investigate and to construct.
2
A rectangle has perimeter 46 cm and one side is 5 cm longer than the other. How long is the shorter side in cm? (algebraic method)
9
3
A right triangle has legs 6 cm and 8 cm and hypotenuse 10 cm. Writing the area in two ways (area method), find the height to the hypotenuse.
S = 6 · 8 / 2 = 24 cm² and S = 10 · h / 2 ⇒ h = 4.8 cm.
4
In proof by contradiction, why does reaching a contradiction prove the theorem?
Of two contradicting statements exactly one is true. The negation of the conclusion led to a contradiction, so it is false, i.e. the conclusion itself is true.