Misc. Write-Up:

Relationships between a Given Triangle

and

Its Medial Triangle

by: BJ Jackson

Ratios of Area and Perimeter for a Triangle

and a Triangle with sides Equal to its Medians

This problem looks at the ratios of area and perimeter of two triangles. Start with a triangle ABC and find the midpoints of each side called EDF. Now, constuct the medians of the triangle. There are now two triangles, the original and triangle EDF whose sides are equal to the length of the medians. Using GSP to find the area and the perimeter of the two triangles, allows one to find the ratios of the areas and the perimeters of the two triangle. The ratio of the areas is 4 and the perimeters is 2. Using GSP to change triangle ABC, shows that the lengths change, but the ratios do not. The pitcure below illustrates the previous claims.

For an interactive GSP sketch, click here.

Note: Although GSP shows that the ratios of the areas and the perimeters appear to be constants, GSP does not prove the claims.

Click here to see a proof that the ratio of the perimeters of triangles ABC and DEF equals 2.

Click here to see a proof that the ratio of the areas of triangles ABC and DEF equals 4.

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