Coordinate Proofs in Geometry

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

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

Example: Using the Midsegment Theorem

^Solution

Fill in the Blanks

$\overline{JK}\ \parallel\ _{ }$

 

Complete the statement.

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)