Goal: Construct a triangle given its three medians t_{a}, t_{b} and t_{c}. 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.)

YouTubeMathcastorScreenCast Mathcast (if YT is blocked)

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?

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·t_{a}, |AD|=2/3·t_{b} and |TD|=2/3·t_{c}.

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·t_{a}+ 2/3·t_{b}£ 2/3·t_{c}. They should see that they can multiply by 3/2 and get the inequality: t_{a}+t_{b}£t_{c}. We have them write down the other 2 analogous inequalities and then have them write the concluding sentence:

A triangle with medians t_{a}+t_{b} and t_{c}exists and is unique if and only if the 3 medians satisfy the triangle inequality.

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Advanced Compass and Straightedge Constructions with GeoGebra

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Construction: Construct a triangle given its medians

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Construct Triangle from Medians## Construct a Triangle Given its 3 Medians

Goal:Construct a triangle given its three medianst,_{a}tand_{b}t. 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.)_{c}YouTubeMathcastorScreenCast Mathcast (if YT is blocked)GeoGebra Exploration:Given 3 medians, create an exploring construction Directions for Exploration Interactivity (opens in new window)GeoGebra Construction:Watch the C+S construction and then test it Directions for InterActivity1. Click on

Playto 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?

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·

t, |AD|=2/3·_{a}tand |TD|=2/3·_{b}t._{c}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·

A triangle with medianst+ 2/3·_{a}t£ 2/3·_{b}t. They should see that they can multiply by 3/2 and get the inequality:_{c}t+_{a}t£_{b}t. We have them write down the other 2 analogous inequalities and then have them write the concluding sentence:_{c}t+_{a}tand_{b}texists and is unique if and only if the 3 medians satisfy the triangle inequality._{c}MetadataRelated themes:

Basic C+S Constructions

Construct Triangle from Mediansgeometric, construct, construction, straightedge, compass, ruler, geogebra, application, geometry, program