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Enlargement from a centre

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

You learn how to enlarge or shrink a figure with a central dilation (zentrische Streckung). The shape stays the same.
📄 Open worksheet (PDF, German)

The video

In the video a triangle is enlarged from the centre \(Z\) with factor \(2\). The compasses carry over the length. Then it is shrunk with \(0.5\).

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

Contents of the video

  1. 0:00 Introduction
  2. 0:16 Stretching a point
  3. 0:46 Stretching the triangle
  4. 1:05 Properties
  5. 1:20 Shrinking
  6. 1:39 Careful, mistake!
  7. 1:58 Try it yourself

The most important points

Central dilation

\(Z\) is the centre (das Streckzentrum), \(k\) the scale factor (der Streckfaktor). The image point \(P'\) lies on the ray from \(Z\) through \(P\), and \(\overline{ZP'} = k \cdot \overline{ZP}\). For \(k > 1\) the figure gets bigger, for \(0 < k < 1\) smaller.

The triangle ABC is stretched with k = 2 from the centre Z.
The triangle \(ABC\) is stretched with \(k = 2\) from the centre \(Z\).

Properties

Every segment becomes \(k\) times as long, the image segments are parallel to the originals, the angles stay the same. The area becomes \(k^2\) times as big.

Shrinking with k = 0.5: every image point lies halfway between Z and the point.
Shrinking with \(k = 0.5\): every image point lies halfway between \(Z\) and the point.
Mistake: the length was measured from A. The point is then three times as far from Z.
Mistake: the length was measured from \(A\). The point is then three times as far 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
Page 1/4
The worksheet – Page 2
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The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

\(\overline{ZP} = 2.5\) cm and \(k = 3\). How far is \(P'\) from \(Z\)?

Show solution

\(\overline{ZP'} = 3 \cdot 2.5 \text{ cm} = 7.5\) cm

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: zentrische Streckung (Streckzentrum, Streckfaktor, Bildpunkt, vergrößern und verkleinern). 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.