# Difference between revisions of "Midpoint"

Twod horse (talk | contribs) |
Twod horse (talk | contribs) |
||

Line 50: | Line 50: | ||

[ABC] &= 4[EFD] | [ABC] &= 4[EFD] | ||

\end{align*}</cmath> | \end{align*}</cmath> | ||

− | == | + | == Cartesian Plane == |

In the Cartesian Plane, the coordinates of the midpoint <math>M</math> can be obtained when the two endpoints <math>A</math>, <math>B</math> of the line segment <math>\overline{AB}</math> is known. Say that <math>A: A(x_A,y_A)</math> and <math>B: B(x_B,y_B)</math>. The Midpoint Formula states that the coordinates of <math>M</math> can be calculated as: | In the Cartesian Plane, the coordinates of the midpoint <math>M</math> can be obtained when the two endpoints <math>A</math>, <math>B</math> of the line segment <math>\overline{AB}</math> is known. Say that <math>A: A(x_A,y_A)</math> and <math>B: B(x_B,y_B)</math>. The Midpoint Formula states that the coordinates of <math>M</math> can be calculated as: | ||

<cmath>\begin{align*} | <cmath>\begin{align*} |

## Revision as of 03:34, 12 February 2021

*This article is a stub. Help us out by expanding it.*

## Definition

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

### 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

## 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: