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Altitude 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 altitude theorem (der Höhensatz): the square on the height is as big as the rectangle made of the two parts of the hypotenuse.
📄 Open worksheet (PDF, German)

The video

In the video a rectangle with sides \(q\) and \(p\) turns into a square of the same area. Then come a number example and the proof with similar triangles.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:18 The altitude theorem
  3. 0:43 A number example
  4. 1:03 Why?
  5. 1:51 Application
  6. 2:13 Careful, mistake!
  7. 2:36 Try it yourself

The most important points

Altitude theorem

In a right-angled triangle the height \(h\) (die Höhe) cuts the hypotenuse into \(p\) and \(q\). Then \(h^2 = p \cdot q\).

The rectangle of q and p becomes a square with side h, with the same area.
The rectangle of \(q\) and \(p\) becomes a square with side \(h\), with the same area.

Proof

The triangles \(ACH\) and \(CBH\) are similar: \(h : q = p : h\). So \(h^2 = p \cdot q\).

Two similar triangles ACH and CBH.
Two similar triangles \(ACH\) and \(CBH\).
Mistake: h^2 = p + q (plus instead of times). Correct is h^2 = p · q.
Mistake: \(h^2 = p + q\) (plus instead of times). Correct is \(h^2 = p \cdot q\).

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

\(p = 2\) and \(q = 8\). How long is the height \(h\)?

Show solution

\(h^2 = p \cdot q = 16\), so \(h = 4\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Höhensatz (Höhe, Hypotenusenabschnitte, Quadrat und Rechteck, ähnliche Dreiecke). 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.