**Assignment 6:**

** Investigating the Relationship between the Area of a Triangle and the Area of the Triangle formed by its Medians**

by

Ángel M. Carreras Jusino

Goal: Prove that the area of a triangle formed by the medians of a given triangle is ¾ the area of the given triangle .

ExplorationIn the following applet is shown

- triangle
ABCwith its medians- triangle
GHI(the triangle formed with the medians of triangleABC)- the areas of
ABCandGHI- the result of multiplying the area of
ABCby ¾.Drag any of the vertices of triangle

ABCand see what happens to the values of the areas and the result of the calculation.

You should note that the area of the triangle of medians is ¾ the area of the original triangle.

Algebraic ProofNow we are going to prove what was noted during the exploration. Consider the following triangle

ABC. In this triangleD,E, andFare the midpoints of the segmentsBC,AC, andABrespectively. ThereforeAD,BE, andCFare the medians of triangleABC.

First, we want to find an expression for the area of triangle

ABC, note that the triangle has base 2x_{1}and height 2y_{2}. Using the area formula for a triangle we have

A= ½(base)(height) = ½(2x_{1})(2y_{2}) = 2x_{1}y_{2}Now we need to find an expresion for the area of the triangle formed by the medians of triangle

ABC, i.e., the triangle with side lengths equal to the lengths of the segmentsAD,BE, andCF. Since we can't determine the area of the triangle of medians as we did with triangleABC, we are going to use Heron's formula to find it.Heron's formula states that the area

Aof a triangle whose sides have lengthsa,b, andciswhere

sis the semiperimeter of the triangle.

This formula can also be written as

.

This version of Heron's formula is what we are going to use for the proof.

Using the distance formula we can find the lengths of the medians, which are going to be the sides of the triangle of medians. So, the area of the triangle of medians is going to be

The lengths of the medians are

Substituing and simplifying for each of the pieces needed for the Heron's formula we get

Therefore the area of the triangle of medians is

We originally have that triangle

ABChas an area of 2x_{1}y_{2}and now we have that the area of the triangle of medians is .∴