Variables, polygons and angles

Animated Pythagoras' Theorem Jigsaw Puzzle. Change the topmost slider to see the translations.

Dynamic software is well suited for finding mathematical patterns. By investing numerical values, it is possible to make conjectures. Such conjectures are of no value if the mathematical reasoning is omitted. It would have been possible to make a program that systematically produced numerical errors. It is important to understand the difference between a demonstration and a proof. In the worksheet above, a proof of the Pythagorean theorem is suggested, an easier example is shown at the page Geometry - Pythagoras & Thales.

Demonstration of Pythagoras' Theorem

A demonstration of Pythagoras' Theorem showing numerical values.

Make a right-angled triangle using these tools:

Icon Line through Two Points
Icon Perpendicular Line
Icon New Point
Icon Polygon

The points of a polygon should be placed in a counterclockwise order!

Use the tool Angle Icon to show the right angle. Move the points! The triangle should remain right-angled.

If you use the tool Angle and then click on a polygon, all angles of the polygon are shown. If the verticies of the polygon have been placed in a counterclockwise order, the interior angles are shown, otherwise the exterior angles are shown.

Image
The triangle to the left is made in a counterclockwise order, the triangle to the right in a clockwise order.

Variables

All objects in GeoGebra have names, so called variable names. You can rename a variable by right-clicking on it in the algebra view and pick Rename.

image

Rename the sides of the triangle to a, b and c; as in the picture below.

image

The sides of the triangle are called a, b and c; the points are called A, B and C; the triangle is called poly1 and the angle α.

A variable has a name and a value. The values of the variables are shown in the algebra view.

  • The values of the points are the coordinates.
  • The values of the sides are the lengths.
  • The value of the triangle is the area.

When moving the points you can see how the values change.

You can change the number of decimals shown by choosing Options->Rounding.

Making variables

In order to demonstrate Pythagoras' theorem we must show the square of the hypotenuse and the sum of the squares of the shorter sides. We hence introduce two variables to store these values.

There is a standard way of writing subscripts and superscripts in GeoGebra; this way of writing is used in a number of mathematics programs.

Superscripts are written using ^     x^2 is shown like this \(x^2\)

Subscripts are written using _     c_1 is shown like this \(x_2\)

 

You write a ^ by pressing Shift+^, the character ^ may not show up until you press the next character or spacebar. You write a _ by pressing Shift+ -.

image

You write variables in the input bar at the bottom of the window. If the input bar is not shown, go to View->Input Bar.

Store the square of the hypotenuse in the variable v1 and the sum of the squares of the shorter sides in v2.

You can observe the values of v1 and v2 as you move the points of the triangle.

Note that:

This is not a proof of Pythagoras' theorem, it is merely a demonstration showing that Pythagoras' theorem possibly could be true!

The demonstration becomes clearer if you introduce text boxes into the drawing pad; this is what the next section is about. Save the right-angled triangle!

A false demonstration

A demonstration implying that Pythagoras' Theorem is false for all triangles but the 3-4-5-triangle.

further info:

more proofs: http://www.cut-the-knot.org/pythagoras/

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se