# The Pythagorean theorem online - examples, interactive exercises

## The Pythagorean Theorem - what does the Pythagorean Theorem say?

**The Pythagorean Theorem - Definition**

In mathematics, the Pythagorean Theorem is an important and fundamental relationship in geometry between the three sides of a right triangle. **The theorem states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.**

Want to learn more about Pythagoras? Who was Pythagoras? When did he live? Just have a look at Wikipedia about Pythagoras!

## The Theorem of Thales - Exercise on right triangle

**Theorem of Thales **

** Thales' theorem** states that if A, B, and C are distinct points on a circle on which the line AB is a diameter, then the angle ACB (at point C) is a right angle.

You are also welcome to try out Thales' theorem in the embedded Geogebra app by moving the point C on the circle with your mouse!

Try it out!

You can also open the Geogebra app directly in Geogebra with this link: https: //www.geogebra.org/calculator/ec7yhuex

## Pythagoras online interactive!

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the two cathets (the sides enclosing the right angle).

You can test this connection interactively here!

**The Pythagorean theorem interactive with GeoGebra**

Move **point B** and/or **point C** and observe how the areas of the right triangles change!

The AB, BC and AC routes are always displayed.

**It is always valid with Pythagoras:**

\(AB^2 + AC^2 = BC^2 \)

**The area of the square above the hypotenuse (red in the app) is as large as the sum of the two areas of the squares above the cathets (green and blue in the app)!**

You can also access this interactive app directly from Geogebra using the following link: https://www.geogebra.org/calculator/qf8tvetb

## Pythagoras explanation with the Geoboard

**Pythagoras online with the Geoboard**

You can also practice the Pythagorean theorem on a geometry board. You can also find a geometry board online via this link:

## Pythagorean theorem - the height theorem

Click on item C in the app to understand the height theorem.

You can also open this interactive app directly in Geogebra with this link: https://www.geogebra.org/calculator/mbh7pwhy

## Pythagorean Theorem - The Cathetus Theorem

**Cathetus set**

The cathetus theorem is a component of the Pythagorean theorem, if the theorem is understood as a group of theorems.

The cathetus theorem states that the square above a cathetus has the same area as the rectangle below the hypotenuse segment on the same side. In our picture, the length of the hypotenuse = AB or c.

- \( b^2 = p\cdot c\) (red!)
- \( a^2 = q\cdot c\) (green!)

Move the **point C** in the app to understand the cathetus theorem. Observe how the area of the squares and rectangles changes.

You can also open the app directly in Geogebra with your link: https://www.geogebra.org/calculator/tzyurrhh

## Pythagorean Theorem Tasks

Task sheets and interactive tasks on the Pythagorean Theorem will follow soon here!