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Intercept theorems

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

You learn the intercept theorems (die Strahlensätze): with two rays and two parallels you calculate lengths, for example the height of a tree.
📄 Open worksheet (PDF, German)

The video

In the video two rays and two parallels make the intercept figure. Then you measure the height of a tree with its shadow.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:12 The intercept figure
  3. 0:32 The first theorem
  4. 0:49 The second theorem
  5. 1:07 The height of the tree
  6. 1:47 Careful, mistake!
  7. 2:09 Try it yourself

The most important points

The intercept theorems

Two rays start at \(Z\), two parallels \(AB \parallel A'B'\) cross them. First theorem: \(\overline{ZA} : \overline{ZA'} = \overline{ZB} : \overline{ZB'}\). Second theorem: \(\overline{ZA} : \overline{ZA'} = \overline{AB} : \overline{A'B'}\).

First theorem: 2 : 5 = 2.4 : 6 = 0.4.
First theorem: \(2 : 5 = 2.4 : 6 = 0.4\).
Second theorem: 2 : 5 = 1.4 : 3.5 = 0.4.
Second theorem: \(2 : 5 = 1.4 : 3.5 = 0.4\).

Tree example: stick \(1.5\) m, shadow \(2\) m, tree shadow \(12\) m. From \(1.5 : 2 = h : 12\) we get \(h = 9\) m.

The height of the tree: stick and tree are parallel, and so are the sun rays.
The height of the tree: stick and tree are parallel, and so are the sun rays.
Mistake: \overline{AA'} taken instead of \overline{ZA'}. Always measure from Z.
Mistake: \(\overline{AA'}\) taken instead of \(\overline{ZA'}\). Always measure from \(Z\).

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
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The worksheet – Page 3
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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

\(\overline{ZA} = 3\), \(\overline{ZA'} = 4.5\), \(\overline{ZB} = 4\). How long is \(\overline{ZB'}\)?

Show solution

\(\overline{ZB'} = 4 \cdot 4.5 : 3 = 6\)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Strahlensätze (Strahlensatzfigur, Verhältnisse, erster und zweiter Strahlensatz, Höhe mit Schatten berechnen). 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.