Midsegment Theorem
of a Triangle and Trapezoid
:
A
midsegment
of a triangle is a segment connecting the midpoints of two of its sides.
Examples:
Midsegment Theorem
–
If a line segment joins the midpoint of two sides of a triangle and
is
parallel to t
he third side, then the length of the line segment is half the length of the
third side.
Strategy:
1) Carefully graph the triangle. Be sure to label your x

axis and y

axis.
2) Use the midpoint formula to calculate parts b and c.
3) Use the distance fo
rmula to calculate the lengths of the two segments in part d.
4) Use the slope formula to prove that the segment joining the two midpoints of
the sides of the triangle is parallel to the third side.
*** For parts 3 & 4, use words and the therefore symbol
to prove your work!!!
Example:
Given: Triangle ABC with vertices A(1,2), B(7,6), and C(7,2).
a) Graph and label triangle ABC.
Solution
: Use the coordinate plane, make sure everything is labeled.
b) Find the coordinates of D, the midpoint of BC.
Sol
ution
: Use the midpoint formula to find the coordinate of D.
c) Find the coordinates of E, the midpoint of AC.
Solution
: Use the midpoint formula to find the coordinates of D.
d) Prove
DE = ½AB.
Solution
: Use the distance formula to find D
E and AB. Put the work for
these two directly next to one another!!! In words explain how the above is true.
e) Prove
that DE is parallel to AB.
Solution
: To prove that DE is parallel to AB, show their slopes are equal.
Put the work for these
two directly next to one another!!!
More Examples:
1) Given triangle ABC with vertices A(

3, 9), B(

5,

3), and C(7,

1).
a) Graph and label triangle ABC.
b) Find the coordinates of D, the midpoint of AB.
c) Find the coordinates of E, the midpoint
of AC.
d) Prove
DE = ½BC.
e) Prove
that DE is parallel to BC.
2) Given triangle ABC with vertices A(

4,8), B(8,12), and C(4,

4).
a) Graph and label triangle ABC.
b) Find the coordinates of D, the midpoint of AB.
c) Find the coordinates of E, the midp
oint of AC.
d) Prove
DE = ½BC.
e) Prove
that DE is parallel to BC.
3) Given triangle ABC with vertices A(

2,4), B(4,6), and C(2,

2).
a) Graph and label triangle ABC.
b) Find the coordinates of D, the midpoint of AB.
c) Find the coordinates of E, the
midpoint of AC.
d) Prove
DE = ½BC.
e) Prove
that DE is parallel to BC.
4) Given triangle PQO with vertices P(4,10), Q(6,0), and O(0,0).
a) Graph and label triangle PQO.
b) Find the coordinates of R, the midpoint of OP.
c) Find the coordinates of S, t
he midpoint of QP.
d) Prove
RS = ½ OQ.
e) Prove
RS is parallel to OQ.
Midsegement of a Trapezoid:
Midsegement of a Trapezoid
–
segment that joins the midpoints of the nonparallel sides
of a trapezoid.
Midsegement Theorem
–
The midsegment of a trapez
oid is 1) parallel to the bases and 2)
half as long as the sum of the lengths of the bases.
5) Given trapezoid ABCD, with vertices A(2,6), B(8,6), C(8,0), and D(0,0).
a) Find the coordinates of M, the midpoint of AD.
b) Find the coordinates of N, the
midpoint of BC.
c) Find the distance of AB.
d) Find the distance of DC.
e) Find the distance of MN.
f)
Prove MN is parallel to AB and CD.
g) Prove MN = ½(AB + CD).
6) Given trapezoid ABCD, with vertices A(3,0), B(0,5), C(10,5), and D(8,0).
a) Find t
he coordinates of M, the midpoint AD.
b) Find the coordinates of N, the midpoint of BC.
c) Find the distance of AB.
d) Find the distance of DC.
e) Find the distance of MN.
f)
Prove MN is parallel to AB and CD.
g) Prove MN = 1/2(AB + CD).
7)
Given tr
apezoid ABCD, with vertices A(

4,

1), B(

4,0), C(

1,3), and D(

2,

1).
a) Find the coordinates of F, the midpoint of AD.
b) Find the coordinates o E, the midpoint of BC.
c) Find the distance of AB.
d) Find the distance of CD.
e) Find the distance of EF
.
g) Prove EF is parallel to AB and CD.
h) Prove EF = ½(AB + CD).
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