Centers of a 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) ?
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.
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