|Perpendicular Bisector Theorem|
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
|Triangle Midsegment Theorem|
If is a midsegment of and
Draw an arbitrary triangle, in the coordinate plane so that lies on the origin, and is horizontal and lies on the -axis.
Since lies on the origin, its coordinates are Point is on the -axis, meaning its -coordinate is The remaining coordinates are unknown, and can be named and Then, the points are Draw midsegment from to By defintion, is the midpoint of and the midpoint of
If the slopes of and are equal, the segments are parallel. Since was drawn as a horizontal line, its slope equals To show that the slope of also equals it can be shown that the -coordinates of its points are the same. To do this, and and the midpoint formula can be used. The -coordinates of and are both It follows that the slope of equals Thus, since and have equal slopes, they are parallel.
Since both and are horizontal, their lengths are given by the difference of the endpoints' -values. The -coordinates of and are and respectively. This gives To find the -coordinates of and the midpoint formula is used. The length of is given by subtracting the -coordinates: Since is half of the midsegment is half the length ofTherefore, a midsegment of a triangle is parallel to the third side of the triangle and half the length.
|Triangle Proportionality Theorem|
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