Midpoint
Contents
Definition
In Euclidean geometry, the midpoint of a line segment is the point on the segment equidistant from both endpoints.
A midpoint bisects the line segment that the midpoint lies on. Because of this property, we say that for any line segment with midpoint
,
. Alternatively, any point
on
such that
is the midpoint of the segment.
Midpoints and Triangles
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Midsegments
As shown in Figure 2, is a triangle with
,
,
midpoints on
,
,
respectively. Connect
,
,
(segments highlighted in green). They are midsegments to their corresponding sides. Using SAS Similarity Postulate, we can see that
and likewise for
and
. Because of this, we know that
Which is the Triangle Midsegment Theorem. Because we have a relationship between these segment lengths,
with similar ratio 2:1. The area ratio is then 4:1; this tells us
Medians
The median of a triangle is defined as one of the three line segments connecting a midpoint to its opposite vertex. As for the case of Figure 2, the medians are ,
, and
, segments highlighted in red.
These three line segments are concurrent at point , which is otherwise known as the centroid. This concurrence can be proven through many ways, one of which involves the most simple usage of Ceva's Theorem and the properties of a midpoint. A median is always within its triangle.
The centroid is one of the points that trisect a median. For a median in any triangle, the ratio of the median's length from vertex to centroid and centroid to the base is always 2:1.
Cartesian Plane
In the Cartesian Plane, the coordinates of the midpoint can be obtained when the two endpoints
,
of the line segment
is known. Say that
and
. The Midpoint Formula states that the coordinates of
can be calculated as:
See Also
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