What is the area of the triangle?

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Discussion Overview

The discussion revolves around a geometry problem involving the area of a triangle defined by the lengths of its medians. Participants explore the relationship between the medians and the area, with a focus on geometric reasoning and calculations.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant presents the problem of finding the area of a triangle given the lengths of its medians: Ma = 3, Mb = 4, Mc = 5.
  • Another participant seeks clarification on the term "median," questioning whether it refers to the lengths of the sides of the triangle.
  • A participant clarifies that the medians are lines from a vertex to the midpoint of the opposite side and suggests that the problem may not be particularly difficult if one knows how to relate areas of triangles to their sides.
  • A further reply proposes a method involving the barycenter and a symmetric point, leading to a calculation that suggests the area of the triangle is 8.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the problem, as there are varying levels of understanding regarding the concept of medians and the calculations involved. The discussion includes both clarification and proposed solutions without agreement on a definitive answer.

Contextual Notes

Some assumptions regarding the properties of medians and their relationship to the area of the triangle may not be fully explored or agreed upon. The calculations presented rely on specific geometric reasoning that may require further validation.

Hessam
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First of all note this is NOT HOMEWORK

This is an interesting geometry problem my old statistics and discrete math teacher gave to me, i solved it, however i just wanted to give it out... very tough problem (at least for me, probably easy for most poeple here)

You have a triangle... the lengths of the three medians are
Ma = 3
Mb = 4
Mc = 5

so what is the area of the triangle?

good luck
 
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Hi!

I'm not a native english speaking person so I don't get the "median" part. Do you mean length of the sides?
 
No, he means the lengths of the medians which are the lines that pass from one corner to the mid point of the opposite sides. It isn't a particularly difficult question if you know how to describe the areas of triangles in terms of sides (at least on first inspection it seems there is an obvious argument to try).
 
Hessam said:
...so what is the area of the triangle?

Let G be the barycenter of ABC, and let P be its symmetric point with respect to middle point of BC.

Then, the triangle BGP is formed by 2/3 of the medians (and it is rectangle).

So, BGParea = (2/3)^2 * (3*4)/2 = 8/3 .

But BGP is 1/3 of ABC.

So, ABCarea is 8.
 
Last edited:

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