The triangle midsegment theorem proof is easy to follow in this lesson. This lesson will give a coordinate proof of the triangle midsegment theorem. What is the triangle midsegment theorem?
If a segment joins the midpoints of the sides of a triangle, then the segment is parallel to the third side and the segment is half the length of the third side.
Proof of the theorem.
Consider the following triangle on the coordinate system.
Given: S is the midpoint of OQ R is the midpoint of PQ Prove: SR  OP
SR =
OP
2

Given: S is the midpoint of OQ R is the midpoint of PQ Prove: SR  OP
SR =
OP
2


S: (
b + 0
2
,

c + 0
2
)

= (
b
2
,

c
2
)

S: (
b + 0
2
,

c + 0
2
)

= (
b
2
,

c
2
)

R: (
a + b
2
,

c + 0
2
)

= (
a + b
2
,

c
2
)

R: (
a + b
2
,

c + 0
2
)

= (
a + b
2
,

c
2
)

To prove that SR is half OP, we can use the distance formula to find SR and OP.
$$ OP = \ {\sqrt{(a  0)^2 + (0  0)^2 } } $$ 
$$ OP = \ {\sqrt{(a)^2 + (0)^2 } } $$ 
$$ OP = \ {\sqrt{(a)^2 } } = a $$ 
$$ SR = \ {\sqrt{(\frac{a+b} {2}  \frac{b} {2} )^2 + (\frac{c} {2} \frac{c} {2} )^2 } } $$ 
$$ SR = \ {\sqrt{(\frac{a} {2}+ \frac{b} {2}  \frac{b} {2} )^2 + (0)^2 } } $$ 
$$ SR = \ {\sqrt{(\frac{a} {2} + 0)^2 + (0)^2 } } $$ 
$$ SR = \ {\sqrt{(\frac{a} {2})^2 } = \frac{a} {2} } $$ 