Name:_______________________
Triangle Midsegment Theorem
Date:_____ Period:____
Ms. Anderle
Triangle Midsegment Theorem
A midsegment of a triangle is a segment whose endpoints are the midpoints of two
sides of the triangle.
Examples:
Triangle Midsegment Theorem: A midsegment of a triangle is parallel to one side of
the triangle, and its length is one-half the length of that side.
Example: If B and D are midpoints of AC and EC respectively, BD || AE and
BD = ½ AE
C
B
A
//
D
//
E
Strategy:
1) Carefully graph the triangle. Be sure to label your x and y axis.
2) Use the midpoint formula to calculate parts b and c.
3) Use the distance formula to calculate the lengths of two segments in
part d.
4) Use the slope formula to prove that the segments joining the two
midpoints of the sides of the triangle is parallel to the third side.
Example:
Given ∆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 AB.
Solution: Use the midpoint formula to find the coordinates of D
c) Find the coordinates of E, the midpoint of BC.
Solution: Use the midpoint formula to find the coordinates of E.
d) Prove DE = ½ AC
Solution: Use the distance formula to find DE and AB. Put the work
for these two directly next to each other! Explain in words how the
above is true.
e) Prove that DE is parallel to AC
Solution: Use the slope formula to show that the slopes of DE and AC
are equal. Put the work directly next to each other!
More Examples: (complete work on separate sheet of paper!!!)
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 midpoint 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, the midpoint of QP.
d) Prove RS = ½ OQ.
e) Prove RS is parallel to OQ.