# Midsegment Theorem

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

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Midsegment Theorem

### Midsegment Theorem

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

The tic marks show that \begin{align*}D\end{align*} and \begin{align*}F\end{align*} are midpoints. \begin{align*}\overline{AD} \cong \overline{DB}\end{align*} and \begin{align*}\overline{BF} \cong \overline{FC}\end{align*}. 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 \begin{align*}\overline{DF}\end{align*} is a midsegment of \begin{align*}\triangle ABC\end{align*}, then \begin{align*}DF = \frac{1}{2} AC = AE = EC\end{align*} and \begin{align*}\overline{DF} \| \overline{AC}\end{align*}.

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 \begin{align*}\triangle FGH \end{align*} and told that \begin{align*} \overline{JK}\end{align*} was its midsegment? How could you find the length of \begin{align*}JK\end{align*} given the length of the triangle's third side, \begin{align*}FH\end{align*}?

### Examples

#### Example 1

Find the value of \begin{align*}x\end{align*} and \begin{align*}AB\end{align*}. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are midpoints.

\begin{align*}AB = 34 \div 2 = 17\end{align*}. To find \begin{align*}x\end{align*}, set \begin{align*}3x - 1\end{align*} equal to 17.

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

#### Example 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.

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 3

The vertices of \begin{align*}\triangle LMN\end{align*} are \begin{align*}L(4, 5), \ M(-2, -7)\end{align*} and \begin{align*}N(-8, 3)\end{align*}. Find the midpoints of all three sides, label them \begin{align*}O, \ P\end{align*} and \begin{align*}Q.\end{align*} Then, graph the triangle, plot the midpoints and draw the midsegments.

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

\begin{align*}L\end{align*} and \begin{align*}M = \left ( \frac{4 + (-2)}{2}, \frac{5 + (-7)}{2} \right ) = (1, -1) \end{align*} point \begin{align*}O\end{align*}

\begin{align*}M\end{align*} and \begin{align*}N = \left ( \frac{-2 + (-8)}{2}, \frac{-7 + 3}{2} \right ) = (-5, -2) \end{align*}, point \begin{align*}P\end{align*}

\begin{align*}L\end{align*} and \begin{align*}N = \left ( \frac{4 + (-8)}{2}, \frac{5 + 3}{2} \right ) = (-2, 4)\end{align*}, point \begin{align*}Q\end{align*}

#### Example 4

Mark all the congruent segments on \begin{align*}\triangle ABC\end{align*} with midpoints \begin{align*}D, \ E\end{align*}, and \begin{align*}F\end{align*}.

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 \begin{align*}\triangle ABC\end{align*}, with midpoints \begin{align*}D, \ E\end{align*}, and \begin{align*}F\end{align*}.

#### Example 5

\begin{align*}M, \ N\end{align*}, and \begin{align*}O\end{align*} are the midpoints of the sides of \begin{align*}\triangle XYZ\end{align*}.

Find \begin{align*}MN\end{align*}\begin{align*}XY\end{align*}, and the perimeter of \begin{align*}\triangle XYZ\end{align*}.

Use the Midsegment Theorem:

\begin{align*}MN = OZ = 5\end{align*}

\begin{align*}XY = 2(ON) = 2 \cdot 4 = 8\end{align*}

Add up the three sides of \begin{align*}\triangle XYZ\end{align*} to find the perimeter.

\begin{align*}XY + YZ + XZ = 2 \cdot 4 + 2 \cdot 3 + 2 \cdot 5 = 8 + 6 + 10 = 24\end{align*}

Remember: No line segment over \begin{align*}MN\end{align*} 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.

\begin{align*}R, \ S, \ T\end{align*}, and \begin{align*}U\end{align*} are midpoints of the sides of \begin{align*}\triangle XPO\end{align*} and \begin{align*}\triangle YPO\end{align*}.

1. If \begin{align*}OP = 12\end{align*}, find \begin{align*}RS\end{align*} and \begin{align*}TU\end{align*}.
2. If \begin{align*}RS = 8\end{align*}, find \begin{align*}TU\end{align*}.
3. If \begin{align*}RS = 2x\end{align*}, and \begin{align*}OP = 20\end{align*}, find \begin{align*}x\end{align*} and \begin{align*}TU\end{align*}.
4. If \begin{align*}OP = 4x\end{align*} and \begin{align*}RS = 6x - 8\end{align*}, find \begin{align*}x\end{align*}.

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

1. The sides of \begin{align*}\triangle XYZ\end{align*} are 26, 38, and 42. \begin{align*}\triangle ABC\end{align*} is formed by joining the midpoints of \begin{align*}\triangle XYZ\end{align*}.
1. What are the lengths of the sides of \begin{align*}\triangle ABC\end{align*}?
2. Find the perimeter of \begin{align*}\triangle ABC\end{align*}.
3. Find the perimeter of \begin{align*}\triangle XYZ\end{align*}.
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 \begin{align*}\triangle ABC\end{align*} below find the midpoints of each side.

1. \begin{align*}A(5, -2), \ B(9, 4)\end{align*} and \begin{align*}C(-3, 8)\end{align*}
2. \begin{align*}A(-10, 1), \ B(4, 11)\end{align*} and \begin{align*}C(0, -7)\end{align*}
3. \begin{align*}A(-1, 3), \ B(5, 7)\end{align*} and \begin{align*}C(9, -5)\end{align*}
4. \begin{align*}A(-4, -15), \ B(2, -1)\end{align*} and \begin{align*}C(-20, 11)\end{align*}

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

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### Vocabulary Language: English Spanish

TermDefinition
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 $@\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)@$.