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Learning Objectives

  • Define midsegment.
  • Use the Midsegment Theorem.

Review Queue

Find the midpoint between the given points.

  1. (-4, 1) and (6, 7)
  2. (5, -3) and (11, 5)
  3. Find the equation of the line between (-2, -3) and (-1, 1).
  4. Find the equation of the line that is parallel to the line from #3 through (2, -7).

Know What? A fractal is a repeated design using the same shape (or shapes) of different sizes. Below, is an example of the first few steps of a fractal. Draw the next figure in the pattern.

Defining Midsegment

Midsegment: A line segment that connects two midpoints of the sides of a triangle.

 \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}

Example 1: Draw the midsegment \overline{DE} between \overline{AB} and \overline{AC} for \triangle ABC above.

Solution: Find the midpoints of \overline{AB} and \overline{AC} using your ruler. Label these points D and E. Connect them to create the midsegment.

Example 2: You now have all three midpoints of \triangle ABC. Draw in midsegment \overline{DF} and \overline{FE}.

Solution:

For every triangle there are three midsegments.

Midsegments in the x-y Plane

Let’s transfer what we know about midpoints in the x-y plane to midsegments in the x-y plane. We will need to use the midpoint formula, \left ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right ) .

Example 3: 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.

Solution: Use the midpoint formula 3 times to find all the midpoints.

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 4: Find the slopes of \overline{NM} and \overline{QO}.

Solution: The slope of \overline{NM} is \frac{-7-3}{-2-(-8)} = \frac{-10}{6} = - \frac{5}{3}. The slope of \overline{QO} is \frac{-1-4}{1-(-2)} = - \frac{5}{3}.

From this we can conclude that \overline{NM} \| \overline{QO}. If we were to find the slopes of the other sides and midsegments, we would find \overline{LM} \| \overline{QP} and \overline{NL} \| \overline{PO}.

Example 5: Find NM and QO.

Solution: Now, we need to find the lengths of \overline{NM} and \overline{QO}. Use the distance formula.

NM &= \sqrt{(-7 -3)^2 + (-2 - (-8))^2} = \sqrt{(-10)^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66\\QO &= \sqrt{(1 - (-2))^2 + (-1 - 4)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83

From this we can conclude that QO is half of NM. If we were to find the lengths of the other sides and midsegments, we would find that OP is half of NL and QP is half of LM.

The Midsegment Theorem

The conclusions drawn in Examples 4 and 5 can be combined into the Midsegment Theorem.

Midsegment Theorem: The midsegment of a triangle is half the length of the side it is parallel to.

If \overline{DF} is a midsegment of \triangle ABC, then DF = \frac{1}{2} AC = AE = EC and \overline{DF} \| \overline{AC}.

Example 6a: Mark all the congruent segments on \triangle ABC with midpoints D, \ E, and F.

Solution: Drawing in all three midsegments, we have:

Also, this means the four triangles are congruent by SSS.

Example 6b: Mark all the parallel lines on \triangle ABC, with midpoints D, \ E, and F.

Solution:

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

Example 7: M, \ N, and O are the midpoints of the sides of the triangle.

Find

a) MN

b) XY

c) The perimeter of \triangle XYZ

Solution: 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.

Example 8: Algebra Connection Find the value of x and AB. A and B are midpoints.

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

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

Know What? Revisited To the left is a picture of the 4^{th} figure in the fractal pattern.

Review Questions

  • Questions 1-5 use the definition of a midsegment and the Midsegment Theorem.
  • Questions 6-9 and 18 are similar to Example 7.
  • Questions 10-17 are similar to Example 8.
  • Questions 19-22 are similar to Example 3.
  • Questions 23-30 are similar to Examples 3, 4, and 5.

Determine if 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. If a line passes through two sides of a triangle and is parallel to the third side, then it is a midsegment.
  5. 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 10-17, 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)

Multi-Step Problem The midpoints of the sides of a triangle are A(1, 5), \ B(4, -2), and C(-5, 1). Answer the following questions. The graph is below.

  1. Find the slope of AB, \ BC, and AC.
  2. The side that passes through A should be parallel to which midsegment? (\triangle ABC are all midsegments of a triangle).
  3. Using your answer from #24, take the slope of \overline{BC} and use the “rise over run” in either direction to create a parallel line to \overline{BC} that passes through A. Extend it with a ruler.
  4. Repeat #24 and #25 with B and C. What are coordinates of the larger triangle?

Multi-Step Problem The midpoints of the sides of \triangle RST are G(0, -2), \ H(9, 1), and I(6, -5). Answer the following questions.

  1. Find the slope of GH, \ HI, and GI.
  2. Plot the three midpoints and connect them to form midsegment triangle, \triangle GHI.
  3. Using the slopes, find the coordinates of the vertices of \triangle RST. (#22 above)
  4. Find GH using the distance formula. Then, find the length of the sides it is parallel to. What should happen?

Review Queue Answers

  1. \left ( \frac{-4+6}{2}, \frac{1+7}{2} \right ) = (1, 4)
  2. \left ( \frac{5+11}{2}, \frac{-3+5}{2} \right ) = (8, 1)
  3. m=\frac{-3-1}{-2-(-1)} = \frac{-4}{-1}=4\!\\y=mx+b\!\\-3=4(-2)+b\!\\b=5, \ y=4x+5
  4. -7=4(2)+b\!\\b=-15, \ y=4x-15

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