Notes for the algebraic derivation of the Triangle Midsegment Theorem:
Given:
ABC has vertices A, B, and C.
The coordinates of A, B, and C are (xA,yA), (xB,yB), and (xC,yC).
D is the midpoint between B and C.
E is the midpoint between A and C.
DE is the midsegment connecting D and E.
Part One:
1.
The coordinates of D are ( ½[xB+xC], ½[yB+yC]).
2.
The coordinates of E are ( ½[xA+xC], ½[yA+yC]).
3.
The slope of segment AB is mAB=(yB-yA)/(xB-xA).
4.
The slope of segment ED is
mED=(½[yA+yC]- ½[yB+yC])/ (½[xA+xC]- ½[xB+xC]).
5.
mED=(½yA - ½yB)/ (½xA- ½xB)= (yA-yB)/(xA-xB).
6.
The slope of the midsegment ED is equal to the
slope of AB (mED=mAB).
7.
Segments ED and AB are parallel.
Part Two:
1.
The x distance between A and B is xB-xA.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
The y distance between A and B is yB-yA.
The coordinates of D are ( ½[xB+xC], ½[yB+yC]).
The coordinates of E are ( ½[xA+xC], ½[yA+yC]).
The x distance between E and D is
½[xA+xC]-½[xB+xC] = ½[xA-xB].
The y distance between E and D is
½[yA+yC]-½[yB+yC] = ½[yA-yB].
Twice the x distance between E and D is xA-xB.
Twice the y distance between E and D is yA-yB.
The x distance between E and D is ½ the
x distance between A and B.
The y distance between E and D is ½ the
y distance between A and B.
The distance between E and D is ½ the distance
between A and B.
Midpoint formula.
Midpoint formula.
Slope formula.
Slope formula.
Distributive property.
Reflexive property.
Definition of parallel
lines.
Distance on a
coordinate grid.
“
Midpoint formula.
Midpoint formula.
Distance on a
coordinate grid.
“
Distributive property.
“
Reflexive property.
“
Distance between two
points.