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Here’s a triangle with midpoints , and . What do you notice? What do you wonder?
Triangle A B C with midpoints M, L and N drawn forming a triangle. Segments A N and C N are marked congruent with two ticks. Segments B L and C L are marked congruent with one tick. Segments A M and B M are marked congruent with three ticks.
Here’s a triangle . Points and are the midpoints of 2 sides.
Here’s a triangle, . is of the way from to . is of the way from to .
What can you say about segment , compared to segment ? Provide a reason for each of your conjectures.
Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If is the midpoint of segment and is the midpoint of segment , then what can we say about and triangle ?
Segment is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if , then . This happens because the entire triangle is a dilation of triangle , with a scale factor of .
In triangle , segment divides segments and proportionally. In other words, =. Again, there is a dilation that takes triangle to triangle , so is parallel to , and we can calculate its length using the same scale factor.