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Leg theorem

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

You learn the leg theorem (der Kathetensatz): the square on a leg is as big as the rectangle made of the hypotenuse and the segment next to it.
📄 Open worksheet (PDF, German)

The video

In the video two squares appear on the legs. They fit exactly two rectangles on the hypotenuse. Then come a number example and the proof.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:21 Height and segments
  3. 0:35 The leg theorem
  4. 0:59 A number example
  5. 1:21 Why?
  6. 2:04 Application
  7. 2:26 Careful, mistake!
  8. 2:54 Try it yourself

The most important points

Leg theorem

In a right-angled triangle: \(a^2 = c \cdot p\) and \(b^2 = c \cdot q\). \(c\) is the hypotenuse (die Hypotenuse), \(p\) and \(q\) are the segments the height (die Höhe) cuts it into.

The square on a is as big as the rectangle made of c and p.
The square on \(a\) is as big as the rectangle made of \(c\) and \(p\).

Proof

The triangles \(ABC\) and \(CBH\) are similar: \(a : c = p : a\). So \(a^2 = c \cdot p\).

Two similar triangles: ABC and CBH have the same angles.
Two similar triangles: \(ABC\) and \(CBH\) have the same angles.
Mistake: a = c · p (square forgotten). Correct is a^2 = c · p.
Mistake: \(a = c \cdot p\) (square forgotten). Correct is \(a^2 = c \cdot p\).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
Page 2/3
Show the worksheet solutions
The worksheet – Page 3
Page 3/3

Try it yourself

Task

\(c = 8\) and \(q = 2\). How long is the leg \(b\)?

Show solution

\(b^2 = c \cdot q = 16\), so \(b = 4\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Kathetensatz (Katheten, Hypotenuse, Höhe, Abschnitte, Quadrat und Rechteck). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Use simple German sentences.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.