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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