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 Construct a Triangle Given its 3 Medians

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Goal:  Construct a triangle given its three medians ta, tb and tc. Base your construction on the definition of the median and the fact that the three medians of a triangle meet at a single point T and T is 2/3 of the way along each of the medians. (T is called the centroid.)

YouTube Mathcast or ScreenCast Mathcast (if YT is blocked)
YT here
SC here
GeoGebra Exploration:  Given 3 medians, create an exploring constructionDirections for Exploration Interactivity (opens in new window)
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GeoGebra Construction: Watch the C+S construction and then test it Directions for InterActivity

1. Click on Play to see the construction unfold.

2. Select the  Move tool and click & drag the endpoints of the three medians. 

3. Drag one of the median endpoints "too" small or "too" big. The triangle disappears. Why?

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Proof 

Now we can ask - when does a triangle exist? We get them to look in detail at Figure 8, that is, at the construction of triangle ATD. This is just a basic C+S construction of a triangle given 3 side lengths. Here the 3 sides have lengths:  |AT|=2/3·ta, |AD|=2/3·tb  and |TD|=2/3·tc.

The students should know that a triangle exists if and only if the sides satisfy the triangle inequality. We write down one inequalities, e.g. 2/3·ta+ 2/3·tb £ 2/3·tc. They should see that they can multiply by 3/2 and get the inequality: ta+ tb £ tc. We have them write down the other 2 analogous inequalities and then have them write the concluding sentence:

A triangle with medians ta+ tb and tcexists and is unique if and only if the 3 medians satisfy the triangle inequality.

Metadata

Global Advanced Compass and Straightedge Constructions with GeoGebra
Brief Construction: Construct a triangle given its medians
Grade 10th grade and up -Geometry
Strand Geometry
Standards CA Geometry 16.0
Keywords construction, straightedge, compass, ruler, geogebra, geometry, angle, triangle, medians, copy
Comments none
Download You can download everything; a zip is in progress ...
Author LFS - contact
Type Freeware - Available for Offline and Online Use - Translatable (html)
Use Requires sunJava player

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