# 4.19: Midsegment Theorem

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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 \(\overline{BC}\).

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.

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 \(\overline{DF} \parallel \overline{AC}\).

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.

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

Example \(\PageIndex{4}\)

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

**Solution**

Drawing in all three midsegments, we have:

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 \(F\).

Example \(\PageIndex{5}\)

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

**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.

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

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

- If \(OP=12\), find \(RS\) and \(TU\).
- If \(RS=8\), find \(TU\).
- If \(RS=2x\), and \(OP=20\), find \(x\) and \(TU\).
- 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.

- The sides of \(\Delta XYZ\) are 26, 38, and 42. \(\Delta ABC\) is formed by joining the midpoints of \(\Delta XYZ\).
- What are the lengths of the sides of \(\Delta ABC\)?
- Find the perimeter of \(\Delta ABC\).
- Find the perimeter of \(\Delta XYZ\).
- 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.

- \(A(5,−2),\: B(9,4)\: and\: C(−3,8)\)
- \(A(−10,1),\: B(4,11)\: and \:C(0,−7)\)
- \(A(−1,3),\: B(5,7)\: and\: C(9,−5)\)
- \(A(−4,−15),\: B(2,−1)\: and\: C(−20,11)\)

## 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