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# 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

What if you were given $\triangle FGH$ 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, $FH$ ? After completing this Concept, you'll be able to use the Midsegment Theorem to solve problems like this one.

### Watch This

First watch this video.

Now watch this video.

### Guidance

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 $\triangle ABC$ , then $DF = \frac{1}{2} AC = AE = EC$ and $\overline{DF} \| \overline{AC}$ .

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. To play with the properties of midsegments, go to http://www.mathopenref.com/trianglemidsegment.html .

#### Example A

The vertices of $\triangle LMN$ 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.

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

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

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

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

The graph is to the right.

#### Example B

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

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 $\triangle ABC$ , with midpoints $D, \ E$ , and $F$ .

#### Example C

$M, \ N$ , and $O$ are the midpoints of the sides of $\triangle XYZ$ .

Find

a) $MN$

b) $XY$

c) The perimeter of $\triangle XYZ$

To solve, use the Midsegment Theorem.

a) $MN = OZ = 5$

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

c) Add up the three sides of $\triangle XYZ$ 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.

### Guided Practice

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

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.

3. Find $y$ . You may assume that the line segment within the triangle is a midsegment.

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

$3x - 1 & = 17\\3x & = 18\\x & =6$

2. 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.

3. Because a midsegment is always half the length of the side it is parallel to, we know that $y=\frac{1}{2}(36)=18$ .

### Practice

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 $\triangle XPO$ and $\triangle YPO$ .

1. If $OP = 12$ , find $RS$ and $TU$ .
2. If $RS = 8$ , find $TU$ .
3. If $RS = 2x$ , and $OP = 20$ , find $x$ and $TU$ .
4. 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.

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

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

### Vocabulary Language: English Spanish

midsegment

midsegment

A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.
Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Midpoint Formula

Midpoint Formula

The midpoint formula says that for endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is $@\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)@$.