`UGA


Kristin Ottofy

Assignment 6

 

Given three line segments j,k, and m. If these are the medians of a triangle, construct the triangle. Show that your construction is correct and that the triangle is unique.

jkm

The medians of a triangle intersect (2/3)rds of the way down, where the centroid is closer to the midpoints of each segment. So, we can trisect each median and let m be our starting median with point C our centroid.

trisectO

 

Now extend the median line m.

 

m extend

Create point P along the extended median line for m that is 1/3 of the length of the median m from the end of the median m that O is closest to.

 

P

Create circle P' that is centered at P and has a radius of 2/3 the length of median k.

 

P'

Now construct circle O' centered at O with radius 2/3 of the length of median j. Label one of the intersections of P' and O' as point A. This is one of the vertices of the triangle.

A

We can now extend segment OA.

OA

We can construct median j by adding 1/2 of the length of OA onto the OA extended line on O away from A.

j

Let A' be the intersection of the extended j median line and the circle O'.

A'

Construct a circle that is centered at A' with radious of 2/3 the length of median m. Construct another circle at O with radius 2/3 of the length of the k segment. Let point S be the intersection of these two circles.

S

Segment SO is 2/3rds the median k but median s is constructible.

median S

Finally, we can connect points A, I, and S to form the triangle.

AIS

To clean it up a bit:

SAI

 

This triangle is unique because if we had chosen any of the other intersections with the circles, we would not have formed a triangle. Thus, only one triangle could have been formed with these medians.

 



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