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Medians and centroid

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

You learn the medians (die Seitenhalbierenden) of a triangle and the centroid (der Schwerpunkt). You also calculate the centroid with coordinates.
📄 Open worksheet (PDF, German)

The video

In the video a cardboard triangle balances on a tip. Then you construct the centroid and calculate with coordinates.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:14 Medians
  3. 0:39 Two to one
  4. 1:01 With coordinates
  5. 1:35 Careful, mistake!
  6. 1:56 Try it yourself

The most important points

Medians and centroid

A median joins a corner with the midpoint of the opposite side. The three medians meet in the centroid \(S\). A cardboard triangle balances on a tip there.

The triangle balances at the centroid.
The triangle balances at the centroid.

Two to one

\(S\) divides every median in the ratio \(2 : 1\): from the corner to \(S\) are two parts, from \(S\) to the midpoint one part. Example: \(9\) cm \(\to\) \(AS = 6\) cm and \(SM_a = 3\) cm.

The three medians.
The three medians.

With coordinates

\(S\left(\frac{x_1 + x_2 + x_3}{3} \Big| \frac{y_1 + y_2 + y_3}{3}\right)\). Example: \(A(0|0)\), \(B(6|0)\), \(C(3|6)\) gives \(S(3|2)\).

The centroid in the coordinate system.
The centroid in the coordinate system.

Careful: Median and altitude are not the same. The median goes to the midpoint of the side, the altitude is perpendicular to the side. Only in an isosceles triangle do they coincide.

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
Page 4/4

Try it yourself

Task

Calculate the centroid of the triangle \(A(2|1)\), \(B(8|1)\), \(C(5|7)\).

Show solution

\(x = \frac{2 + 8 + 5}{3} = 5\), \(y = \frac{1 + 1 + 7}{3} = 3\). \(S(5|3)\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Seitenhalbierende und Schwerpunkt (2 : 1, Koordinaten). 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.