# Difference between revisions of "Midpoint"

Twod horse (talk | contribs) |
Twod horse (talk | contribs) |
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Line 12: | Line 12: | ||

dot((2,0)); | dot((2,0)); | ||

label("M",(2,0),N); | label("M",(2,0),N); | ||

+ | label("Figure 1",(2,0),4S); | ||

</asy> | </asy> | ||

+ | == Midpoints and Triangles == | ||

+ | <asy> | ||

+ | 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); | ||

+ | </asy> | ||

+ | === Midsegments === | ||

+ | As shown in Figure 2, <math>\Delta ABC</math> is a triangle with <math>D</math>, <math>E</math>, <math>F</math> midpoints on <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CA}</math> respectively. Connect <math>\overline{EF}</math>, <math>\overline{FD}</math>, <math>\overline{DE}</math> (segments highlighted in green). They are midsegments to their corresponding sides. Using SAS Similarity Postulate, we can see that <math>\Delta CFE \sim \Delta CAB</math> and likewise for <math>\Delta ADF</math> and <math>\Delta BED</math>. Because of this, we know that | ||

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

+ | AB &= 2FE \\ | ||

+ | BC &= 2DE \\ | ||

+ | CA &= 2ED \\ | ||

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

+ | Which is the Triangle Midsegment Theorem. Because we have a relationship between these segment lengths, <math>\Delta ABC \sim \Delta EFD (SSS)</math> with similar ratio 2:1. The area ratio is then 4:1; this tells us | ||

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

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

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

== In Cartesian Plane == | == In 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: |

## Revision as of 03:31, 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

## In 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: