Difference between revisions of "Midpoint"
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== Definition == | == Definition == | ||
− | + | In '''Euclidean geometry''', the '''midpoint''' of a [[line segment]] is the [[point]] on the segment equidistant from both endpoints. | |
A midpoint [[bisect]]s the line segment that the midpoint lies on. Because of this property, we say that for any line segment <math>\overline{AB}</math> with midpoint <math>M</math>, <math>AM=BM=\frac{1}{2}AB</math>. Alternatively, any point <math>M</math> on <math>\overline{AB}</math> such that <math>AM=BM</math> is the midpoint of the segment. | A midpoint [[bisect]]s the line segment that the midpoint lies on. Because of this property, we say that for any line segment <math>\overline{AB}</math> with midpoint <math>M</math>, <math>AM=BM=\frac{1}{2}AB</math>. Alternatively, any point <math>M</math> on <math>\overline{AB}</math> such that <math>AM=BM</math> is the midpoint of the segment. | ||
Line 17: | Line 17: | ||
pair A,B,C,D,E,F,G; | pair A,B,C,D,E,F,G; | ||
A=(0,0); | A=(0,0); | ||
− | B=(4,0 | + | B=(4,0; |
− | C=(1,3) | + | C=(1,3) |
D=(2,0); | D=(2,0); | ||
E=(2.5,1.5); | E=(2.5,1.5); |
Revision as of 22:00, 24 February 2021
Contents
[hide]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
pair A,B,C,D,E,F,G; A=(0,0); B=(4,0; C=(1,3) D=(2,0); E=(2.5,1.5); F=(0.5,1.5); G=(5/3,1); draw(A--B--C--cycle); draw(D--E--F--cycle,green); dot(A--B--C--D--E--F--G); draw(A--E,red); draw(B--F,red); draw(C--D,red); label("A",A,S); label("B",B,S); label("C",C,N); label("D",D,S); label("E",E,E); label("F",F,W); label("G",G,NE); label("Figure 2",D,4S); (Error making remote request. Unknown error_msg)
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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