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Midsegment Theorem

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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

CK-12 Midsegment Theorem

First watch this video.

James Sousa: Introduction to the Midsegments of a Triangle

Now watch this video.

James Sousa: Determining Unknown Values Using Properties of the Midsegments of a Triangle

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.

CK-12 Midsegment Theorem

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.

Answers:

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)

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