The Triangle Midsegment Theorem

Midsegment Theorem and Coordinate Proof

Geometry

Mrs. LaGrange

Math spoken here!

Midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has three (3) midsegments.

Finding Midsegments:

Note that point D, E, and F are midpoints of their respective segments.

If we connect each midpoint by line segments, we obtain all three midpoint segments.

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.
Notice, by the midsegment theorem,

 $\overline{DE}\parallel\overline{BC},\$  and  $DE\ =\frac{1}{2}BC$ . 

Example: Length of MidsegmentsNote that A and B are midpoints. By the Midsegments Theorem we know that segment AB  is 1/2 the length of segment XZ. 

 $AB\ =\ \frac{1}{2}XZ$ 

  $3x-1\ =\frac{1}{2}\left(34\right)$  
 $3x\ -1\ =17$  
 $3x\ =\ 18\ \ \rightarrow\ x\ =\ 6\ and\ AB=17.$  

Multiple Choice

Find the length of the segment.

• 5

• 10

• 5/2

Multiple Choice

Find x.

• 2x-68

• 68

• 34

• x-34

Multiple Choice

Find v.

• 33

• 66

• 2v+66

• v+33

Multiple Choice

Challenge problem: Find t.

• 35

• 40

• 5

• 4.5

Real World Example

Roof Trusses

Multiple Choice

Look at the diagram from the previous example. We were given two midsegments,

$$\overline{UV}$$  and  $$\overline{VW}$$ , name the third midsegment of  $$\Delta RST$$  .  (Note that it may or may not be represented with a brace.)

• $$\overline{SU}$$  

• $$\overline{RW}$$  

• $$\overline{UW}$$  

Multiple Choice

Suppose UW = 81, find ST.

• 81

• 162

• 40.5

Example: Using the Midsegment Theorem

 In the kaleidoscope image, $\overline{AE}\cong\overline{BE}$ and  $\overline{AD}\cong\overline{CD}.$  Show that  $\overline{CB}\parallel\overline{DE}.$  

^Solution

Because $\overline{AE}\cong\overline{BE}$ and  $\overline{AD}\cong\overline{CD,}\ E$ is the midpoint of  $\overline{AB}$ and $D$ is the midpoint of  $\overline{AC}$  by definition.  Then  $\overline{DE}$  is a midsegment of   $\Delta ABC$  by definition and  $\overline{CB}\parallel\overline{DE}$  by the Midsegment Theorem.   

Fill in the Blanks

Coming up:

You Try!

You will need a piece of paper, a pencil, and a calculator for the following questions - similar to the previous example.

Multiple Choice

Find the coordinates of D, E, and F.

• D(-2, -4), E(0, -2), F(-4, -1)

• D(-4, -2), E(-2, 0), F(-1, -4)

• D(4, -2), E(2, 0), F(1, -4)

Multiple Choice

Find the slope of $$\overline{DE}$$ .   Recall: Slope (m) = $$\frac{y_2-y_1}{x_2-x_1}$$ . 

• -1

• 1

• 1/2

• -1/2

Multiple Choice

Find the slope of $$\overline{BC}$$ . 

• -1

• 1

• 1/2

• -1/2

Multiple Choice

Recall that parallel lines/segments/rays have the same slope. Are the segments, $$\overline{DE}\ and\ \overline{CB}$$ parallel? 

• yes

• no

Multiple Choice

Use the distance formula to find the length of $$\overline{DE}$$ .  Recall the distance formula:   $$z\ =\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$  

• 4

• 2

• $$2\sqrt{2}$$  

• 16

Multiple Choice

Use the distance formula to find the length of $$\overline{BC}$$ .  Recall the distance formula:   $$z\ =\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$  

• $$4\sqrt{2}$$  
  

• 32

• $$2\sqrt{2}$$  

• $$\sqrt{8}$$  

Multiple Choice

True or False: $$\overline{DE}=\frac{1}{2}\overline{CB}.$$  

• True

• False

Great Job!

You have used midsegments in the coordinate plane to show that two segments are parallel and that one is half the length of the other. Proving (by example) the Midsegment Theorem.

Extra Practice: IXL Geometry M1 (50 IXL points for full credit)