Perpendicular Bisectors of a Triangle

Transcription

1 Perpendicular Bisectors of a Triangle Constructions using Sketchometry Draw three line segments and the corresponding perpendicular bisectors. Hang the three line segments together build an open traverse line. Explorations Look at the perpendicular bisectors while closing the traverse line (building a triangle). Write down your observation. Draw a free circle and try to drag until the circumference meets the three vertices of the triangle. Where is the midpoint of the circle now? Write down a result. Can you find a mathematical argumentation? Check at a new sheet: Draw a triangle with its perpendicular bisectors. Drag the vertices and watch the intersection point of the perpendicular bisectors. Draw the circumference.

2 Perpendicular Bisectors of a Triangle Tips for Teachers: Students should know the term perpendicular bisector. Using a suitable video students are able to learn how to draw line segments find midpoints or using the snap to point mode. Students should draw perpendicular bisectors of three free line segments on their own. For the first the traverse line is open and the three perpendicular bisectors are three lines in general position. These three lines (mostly) do not intersect. As soon as the traverse line is closed and you see a triangle, the three perpendicular bisectors intersect in one point (experimental result). In addition the students can realise that the common intersection point is the midpoint of the circumcircle of the triangle. The teacher may give mathematical argumentations for the common intersection point and the midpoint of the circumcircle. Instructions for Students: Constructions using Sketchometry Draw three line segments and the corresponding perpendicular bisectors. Hang the three line segments together build an open traverse line. Explorations Look at the perpendicular bisectors while closing the traverse line (building a triangle). Note your observation. Draw a free circle and try to drag until the circumference meets the three vertices of the triangle. Where is the midpoint of the circle now? Note a result. Can you find a mathematical argumentation? Check at a new sheet: Draw a triangle with its perpendicular bisectors. Drag at the vertices, watch the intersection point of the perpendicular bisectors and draw the circumference.

3 Theorem of Pythagoras: Experiments Constructions using Sketchometry: Draw a rectangular triangle ABD with [AB] as a hypotenuse Explorations: Measuring... What strikes you?... Conjecture... Now: Let D be a free point...drag D outside or inside of the circle...write down your observations!

4 Theorem of Pythagoras: Proof Constructions using Sketchometry: Draw a rectangular triangle ABD (using the theorem of Thales)... EFG is a triangle which is congruent to ABD. Tip:... Squares... Complete both figures... a+b ( = AD+BD ). Tip:... half-lines, perpendicular lines Mathematical argumentation: Side-length of the right square... Compare Drag point E to point A...

5 The Indian and the River Problem: An Indian is at point I. His tent Z is on the other side of a linear river b. Search for the shortest connection from I to Z, but pay attention: the Indian has to walk a given length s in the river in order to hide his track. Construction using Sketchometry: Draw the points and the river.. The Indian may reach b at point F... Draw s (s is a fixed line segment). Copy s Explorations:... Measure the length of the segments Support:... Mathematical argumentation:... Give a description of the construction ( using only circle and ruler )... Quelle der Aufgabe: Krauter/Bescherer, Erlebnis Elementargeometrie, Kap. 2.2

6 Angle Bisectors in a Triangle Constructions using Sketchometry Draw a triangle ABC. Draw a line p parallel to AB through C. Lay the line segments a (= segment BC ) and b (=segment AC ) on p. They both meet at point C. Connect the free vertices D and E with B an A respectively. Point of intersection is F. Explorations Measuring of the angles FAC and BAF. Analogous measuring at point B.... Role of point F?

7 Divisions of an Angle Problem: The red segments are of the same length. Determine the size of the angle α! Constructions using Sketchometry: Mathematical Arguments:... Variations: Let β be an arbitrary angle ( 90 o ). New construction... New calculation. A similar construction::

8 Parallelograms Inside of a Rectangle Problem: Observe parallelograms inscribed in a rectangle that have sides parallel to the diagonals of the rectangle. Constructions using Sketchometry: Draw... Explorations: Circumferences of the parallelograms. Conjecture:... Support:... Give mathematical arguments:...

9 Equilateral Triangles Problem: You see two equilateral triangles ABC and ADE with common vertex A. F and G are the midpoints of the segments [BD] and [EC]. Watch the triangle AFG while dragging the free points. Construction using Sketchometry: Draw... Explorations: Mathematical arguments:...

10 Chords of Equal Length Problem: You have two circles which intersect in two points (A an B). Construct a line g with A g but B g, so that the two chords generated by g have equal length. Constructions using Sketchometry: Draw two circles which intersect in two points A and B Mathematical Arguments: