Problem
Construct a triangle and its medians. Construct a
second triangle with the three sides having the lengths of the three medians
from your first triangle. Find some relationship between the two triangles.
(E.g., are they congruent? similar? Have same area? same perimeter? ratio of
areas? ratio or perimeters?) Prove whatever you find.
Overview
Relationships between the segments and angles formed
in the construction of a triangle and its medial triangle are investigated.
This problem was explored using the Key Curriculum Press, Geometer’s Sketchpad
® software. Questions regarding the ratio of areas and perimeters as well as
similarity and congruency of angles, sides and triangles are explored using the
software. Observations are made and hypotheses tested, within the exploration,
and a proof of each result is offered below.
Investigation
Upon construction of









Table 6.1
From these measurements it is clear that, for this
specific triangle at least, congruence relationships exist. Further, we
hypothesize congruence relationships for angles. To investigate, it is
necessary only to select three points, which together represent an angle
(e.g. D, B and F for












Table 6.2
An understanding of congruent triangles should be
developed from this exploration. We may at this time conjecture that



Table 6.3
What we notice, however is that the ratio of side
lengths is 2:1 and this is also verified in Sketchpad® by selecting two
corresponding segments, such as
The question that arises is whether any of our
hypotheses are true in general, or just for this triangle. Again, Sketchpad®
may be used to investigate, by either constructing different triangles (and
repeating our previous steps) or by dynamically changing the properties of our
original triangle, while measuring each of the quantities mentioned. A GSP file is provided as an illustration. However, while
we have verified the abovementioned properties for more triangles than just
our original, we have not yet proven this in general, nor have we addressed the
issues of perimeter or area.
Note to self: come back and fix
fonts.
Proof:
Given
By the SSS Theorem (Side, Side, Side), we have
The Definition of Similar Polygons states that two
polygons are similar if and only if their corresponding angles are congruent
and the measures of their corresponding sides are proportional. Therefore,
since
Proof:
Part 1:
Since
Part 2:
Part 3: