### Midsegments of Triangles

```M IDSEGMENTS OF T RIANGLES
Honors Geometry
Vocabulary
Midsegment of a triangle – a
segment connecting the
midpoints of two sides of the
triangle.
Midsegment
Investigation
#
Length & slope of
midsegment
Slope Length
Length & slope of
triangle side
Slope Length
1
2
3
Geogebra Exploration
Theorem
Triangle Midsegment Theorem- If
a segment joins the midpoints of
two sides of a triangle, then the
segment is parallel to the third
side, and is half as long.
Vocabulary
Coordinate Proof – Using
coordinate geometry and algebra
to prove a hypothesis.
We can use a coordinate proof to
prove the Triangle Midsegment
Theorem.
Coordinate Proof of
the Triangle
Midsegment Theorem
Step 1: Plot 3 points on the coordinate
grid. Label them A, B, & C. Connect the
points to form ABC.
A(6,6), B(4,-6) & C(-8,2)
Coordinate Proof of
the Triangle
Midsegment Theorem
Step 2: Algebraically determine the
midpoint of sides AB and BC. Then plot
those points. Label them D & E.
Midpoint: x1+x2 y1+y2
2 , 2
Midpoint AB: 6+4 6+-6
2 , 2
D(5,0) &
E(-2,-2)
Coordinate Proof of
the Triangle
Midsegment Theorem
Step 3: Calculate the slopes of AC & DE.
rise
Slope =
run
mAC = 2/7
mDE = 2/7
Coordinate Proof of
the Triangle
Midsegment Theorem
Step 4: Calculate the lengths of
AC & DE.
d =  (x2 – x1)2 + (y2 – y1) 2
AC = 212 = 253
DE = 53
Applying theTriangle
Midsegment Theorem
A
E
D
B
AB = 10
and
CD = 18.
Find EB,
BC, and
AC.
C
Applying theTriangle
Midsegment Theorem
X
65
Y
Find
mVUZ.
U
V
Z
Applying theTriangle
Midsegment Theorem
BC, and
DC.
A
x + 50
E
x
D
x + 85
3x + 46
B
C
Applying theTriangle
Midsegment Theorem
A
B
F 70
C
140
E