Suppose is the perpendicular bisector of and that is the midpoint of
Two triangles can be created by connecting points and and and
These triangles both have a right angle and one of the legs measures half of They also share one leg,
Therefore, any point on a perpendicular bisector is equidistant from the endpoints of the segment. This can be summarized in a two-column proof.
|SAS congruence theorem|
|Definition of congruent segments|
Determine the perimeter of
A midsegment is a segment that connects the midpoints of two of the sides of a triangle.
Since a triangle has three sides, there are three midsegments.
The segment that connects the midpoints of two sides of a triangle — a midsegment — is parallel to the third side of the triangle and half the length.
If is a midsegment of and
Based on the diagram, the following relation holds true.
A median of a triangle is a line segment between the midpoint of one side and the opposite vertex.
Since a triangle has three vertices, every triangle has three medians.
The three medians of a triangle intersect at one point called the centroid.
Consider The points and are midpoints on their respective side. Thus, and are medians.
The two medians intersect at the point
Now, two new points are introduced — the midpoints of and Call them and
Since and are midpoints of and is a midsegment of Thus, by the Triangle Midsegment Theorem, is parallel to and half the length of
Similarly, is a midsegment of since and are the midpoints of and Therefore, is also parallel to and half the length of It follows that
Thus, the median intersects at two-thirds of the distance from Now, by applying the same reasoning for the third median, it also intersects at two-thirds from
The median from point also intersects at two-thirds from point Therefore, the centroid of a triangle is two-thirds the distance from each vertex to the midpoint of the opposite side.
Determine the length of