Anna Fornaciari, Carlotta D’Itria and Flavia Pavesi from 2LT class, supported by Prof. Padrini, present a graphical and theoretical report about geometry with which they have solved a problem using one of the most useful theorems of geometry studied during Maths classes.
Problem of the carpenter
“A carpenter needs to divide a wooden rod into equal parts only by using a compass and an unmarked ruler.”
- Exercise: Dividing a line segment into 7 equal parts by applying Thales’ theorem.
- Program used: GeoGebra
By sketching a point on the plane, the student draws a half-line called s which corresponds to the unmarked ruler with its initial point called O. From point O he draws a line segment OB which is not parallel to s and corresponds to the wooden rod. By using a compass and adjusting the width as desired, he draws a circumference called c with its centre in O in the half-line and then creates 6 other circumferences congruent to c with the centre in the intersection between the half-line and the circumferences. From the circumferences, he obtains 7 congruent segments which are congruent to the rays of the circumferences.
OQ = QT = TR = RM = MN = NI = IS
The student draws a line called z from point I and makes z drop as a perpendicular line in segment OB.Then the student draws a bundle of 6 lines parallel to BI passing through the centre of the circumferences and gives them the following letters: g, f, p, L h, k. Line g passes through point O, f through C and Q, p through D and T, j through E and R, h through F and M, k through G and N and z through B and I. The student then obtains congruent segments on OB.
OC = CD = DE = EF = FG= GB
The construction is correct because it is an application of Thales’ little theorem.
By using Thales’ theorem, the student states that the segments obtained on OB (the wooden rod) are congruent, so it is possible to divide a given line segment into equal parts without using any subdivisions in units of measurement.
By moving the first circumference, every circumference changes its width but remains congruent to the other. The line segments formed on OB cannot vary in length because they depend on the parallel lines and on OB. By moving OB, the length varies but the parts remain congruent.
To visualize the geometrical construction, click on the link :
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