The logic of planimetry and problem-solving methods
Answers are for parents and teachers.
1
Write the hypothesis and conclusion of the theorem “if a quadrilateral is a parallelogram, its opposite sides are equal”.
Hypothesis: the quadrilateral is a parallelogram. Conclusion: its opposite sides are equal.
2
A right triangle has one leg of 8 cm, and its hypotenuse is 4 cm longer than the other leg. Find its sides by the algebraic method.
Other leg x: x² + 64 = (x + 4)² ⇒ 8x = 48 ⇒ x = 6; the sides are 8 cm, 6 cm, 10 cm.
3
Prove by the coordinate method: the diagonals of a rectangle bisect each other (A(0, 0), B(a, 0), C(a, b), D(0, b)).
The midpoint of AC is (a/2, b/2) and the midpoint of BD is also (a/2, b/2); the midpoints coincide.
4
A triangle with sides 13, 13 and 10 cm has area 60 cm² by Heron’s formula. How long is the height to the 10 cm side, in cm?
12
5
Explain the difference between the synthetic and analytic methods in one sentence.
In the synthetic method we go from the data to the conclusion; in the analytic method we start from the conclusion and work backwards to what is needed to prove it.