4.19: Midsegment Theorem

Midsegment of a triangle joins the midpoints of two sides and is half the length of the side it is parallel to.

A line segment that connects two midpoints of the sides of a triangle is called a midsegment. $\overline{DF}$ is the midsegment between $\overline{AB}$ and $\overline{BC}$.

Figure $\PageIndex{1}$

The tic marks show that $D$ and $F$ are midpoints. $\overline{AD}\cong \overline{DB}$ and $\overline{BF}\cong \overline{FC}$. For every triangle there are three midsegments.

Figure $\PageIndex{2}$

There are two important properties of midsegments that combine to make the Midsegment Theorem. The Midsegment Theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side. So, if $\overline{DF}$ is a midsegment of $\Delta ABC$, then $DF=\dfrac{1}{2}AC=AE=EC$ and $\overline{DF} \parallel \overline{AC}$.

Figure $\PageIndex{3}$

Note that there are two important ideas here. One is that the midsegment is parallel to a side of the triangle. The other is that the midsegment is always half the length of this side.

What if you were given $\Delta FGH$ and told that $\overline{JK}$ was its midsegment? How could you find the length of $JK$ given the length of the triangle's third side, $FH$?

Example $\PageIndex{1}$

Find the value of $x$ and AB. $A$ and $B$ are midpoints.

Figure $\PageIndex{4}$

Solution

$AB=34\div 2=17$. To find $x$, set $3x−1$ equal to 17.

$\begin{align*} 3x−1&=17 \\ 3x&=18 \\ x&=6\end{align*}$

Example $\PageIndex{2}$

True or false: If a line passes through two sides of a triangle and is parallel to the third side, then it is a midsegment.

Solution

This statement is false. A line that passes through two sides of a triangle is only a midsegment if it passes through the midpoints of the two sides of the triangle.

Example $\PageIndex{3}$

The vertices of $\Delta LMN$ are $L(4,5),\: M(−2,−7)\:and\: N(−8,3)$. Find the midpoints of all three sides, label them O, P and Q. Then, graph the triangle, plot the midpoints and draw the midsegments.

Solution

To solve this problem, use the midpoint formula 3 times to find all the midpoints. Recall that the midpoint formula is $\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$.

$L$ and $M=\left(\dfrac{4+(−2)}{2}, \dfrac{5+(−7)}{2}\right)=(1,−1),\: point\: O$

$M$ and $N=\left(\dfrac{−2+(−8)}{2},\dfrac{−7+3}{2}\right)=(−5,−2),\: point\: P$

$L$ and $N=\left(\dfrac{4+(−8)}{2}, \dfrac{5+3}{2}\right)=(−2,4),\: point\: Q$

Figure $\PageIndex{5}$

Example $\PageIndex{4}$

Figure $\PageIndex{6}$

Mark all the congruent segments on $\Delta ABC$ with midpoints $D$, $E$, and $F$.

Solution

Drawing in all three midsegments, we have:

Figure $\PageIndex{7}$

Also, this means the four smaller triangles are congruent by SSS.

Now, mark all the parallel lines on $\Delta ABC$, with midpoints $D$, $E$, and $F$.

Figure $\PageIndex{8}$

Example $\PageIndex{5}$

$M$, $N$, and $O$ are the midpoints of the sides of $\Delta \(x$YZ\).

Figure $\PageIndex{9}$

Solution

Find $MN$, $XY$, and the perimeter of $\Delta \(x$YZ\).

Use the Midsegment Theorem:

$MN=OZ=5$

$XY=2(ON)=2\cdot 4=8$

Add up the three sides of $\Delta XYZ$ to find the perimeter.

$XY+YZ+XZ=2\cdot 4+2\cdot 3+2\cdot 5=8+6+10=24$

Remember: No line segment over MN means length or distance.

Review

Determine whether each statement is true or false.

1. The endpoints of a midsegment are midpoints.
2. A midsegment is parallel to the side of the triangle that it does not intersect.
3. There are three congruent triangles formed by the midsegments and sides of a triangle.
4. There are three midsegments in every triangle.

R, S, T, and U are midpoints of the sides of $\Delta XPO$ and $\Delta YPO$

Figure $\PageIndex{10}$

5. If $OP=12$, find $RS$ and $TU$.
6. If $RS=8$, find $TU$.
7. If $RS=2x$, and $OP=20$, find $x$ and $TU$.
8. If $OP=4x$ and $RS=6x−8$, find $x$.

For questions 9-15, find the indicated variable(s). You may assume that all line segments within a triangle are midsegments.

9. 

   Figure $\PageIndex{11}$

10. 

    Figure $\PageIndex{12}$

11. 

    Figure $\PageIndex{13}$

12. 

    Figure $\PageIndex{14}$

13. 

    Figure $\PageIndex{15}$

14. 

    Figure $\PageIndex{16}$

15. 

    Figure $\PageIndex{17}$

16. The sides of $\Delta XYZ$ are 26, 38, and 42. $\Delta ABC$ is formed by joining the midpoints of $\Delta XYZ$.
    1. What are the lengths of the sides of $\Delta ABC$?
    2. Find the perimeter of $\Delta ABC$.
    3. Find the perimeter of $\Delta XYZ$.
    4. What is the relationship between the perimeter of a triangle and the perimeter of the triangle formed by connecting its midpoints?

Coordinate Geometry Given the vertices of $\Delta ABC$ below find the midpoints of each side.

17. $A(5,−2),\: B(9,4)\: and\: C(−3,8)$
18. $A(−10,1),\: B(4,11)\: and \:C(0,−7)$
19. $A(−1,3),\: B(5,7)\: and\: C(9,−5)$
20. $A(−4,−15),\: B(2,−1)\: and\: C(−20,11)$

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.1.

Resources

Vocabulary

Term	Definition
midsegment	A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.
Congruent	Congruent figures are identical in size, shape and measure.
Midpoint Formula	The midpoint formula says that for endpoints $(x_1,y_1)$ and $(x_2,y_2)$, the midpoint is (\dfrac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).

Additional Resources

Video: Determining Unknown Values Using Properties of the Midsegments of a Triangle

Activities: Midsegment Theorem Discussion Questions

Study Aids: Bisectors, Medians, Altitudes Study Guide

Practice: Midsegment Theorem

Real World: Midsegment Theorem