Centers of a Triangle:

Medial Triangle

By Brooke Norman

For this assignment, I am going to construct a triangle.  Using this triangle, I am going to create its Medial triangle.

I began by using the GSP program to construct a triangle.  I labeled the 3 points A, B, and C.  Take notice to the different colors of the line segments, AB, BC, and CA.

Then, I constructed the midpoints of the 3 sides of triangle ABC.  D is the midpoint of AB, E is the midpoint of BC, and F is the midpoint of CA.  I then connected the points to make a triangle.  This is referred to as the MEDIAL triangle.  The Medial triangle is 1/4 of the area of the original triangle, ABC.  The lengths of the sides in the medial triangle are 1/2 of the length of their corresponding side in the original triangle, ABC.

Now that the Medial triangle is constructed, lets see if it fits the definition of a medial triangle.  Is the area really 1/4 of the original triangle?  Are the length of the sides really 1/2 of that of the original triangle?

I first found the heights of each triangle.  For triangle ABC, I took the perpendicular line to point A and line segment BC.  For triangle EFD, I took the perpendicular line to point E and line segment FD.  I then found the heights by measuring the lengths of AG and HE.  I also found the lengths of all of the sides of both the original triangle, ABC; and the Medial triangle, EFD.

Remember the formula for the area of a triangle is A=1/2 Base * Height.

I will now calculate the area of triangle ABC.

A(ABC)= 1/2 (CB)(AG)= 1/2 (11.85)(6.42)=38.0385 cm2

A(EFD)= 1/2 (DF)(HE)= 1/2 (5.93)(3.21)=9.51765 cm2

Is A(EFD = 1/4 A(ABC) ?

38.0385/4= 9.509625

Are they the same?  Pretty close, they both equal 9.51 cm2

Lastly, I am going to compare the side lengths of the original triangle, ABC; to that of the medial triangle, EFD.  In order to help you see the corresponding sides, I labeled them the same colors.

Compare DE & AC.  Is 5.31= 1/2 10.63?   YES!!

Compare DF & CB.  Is 5.93= 1/2 11.85?  YES!!

Compare FE & BA.  Is 3.63= 1/2 7.26?  YES!!

I can say that triangle EFD is indeed the Medial triangle of triangle ABC.  I found that all of the sides of the medial triangle are equal to 11/2 of the lengths of the sides of the original triangle, ABC.  I also found that the area of the Medial triangle, EFD is 1/4  the area of the original triangle, ABC.