Variables, polygons and angles

Demonstration of Pythagoras' Theorem

Make a right-angled triangle using these tools:

Line through Two Points
Perpendicular Line
New Point
Polygon

Use the tool Angle 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.

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.

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

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.

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+ -.

You write variables in the input bar at the bottom of the window.

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!

There is a proof on the page Geometry - Pythagoras' Theorem.

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!