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# 5.1: Midsegments

Difficulty Level: At Grade Created by: CK-12

## 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.

DF¯¯¯¯¯¯¯¯\begin{align*} \overline{DF}\end{align*} is the midsegment between \begin{align*} \overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*}.

The tic marks show that \begin{align*}D\end{align*} and \begin{align*}F\end{align*} are midpoints.

\begin{align*}\overline{AD} \cong \overline{DB}\end{align*} and \begin{align*}\overline{BF} \cong \overline{FC}\end{align*}

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

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

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

Solution:

For every triangle there are three midsegments.

## Midsegments in the \begin{align*}x-y\end{align*} Plane

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

Example 3: The vertices of \begin{align*}\triangle LMN\end{align*} are \begin{align*}L(4, 5), \ M(-2, -7)\end{align*} and \begin{align*}N(-8, 3)\end{align*}. Find the midpoints of all three sides, label them \begin{align*}O, \ P\end{align*} and \begin{align*}Q\end{align*}. Then, graph the triangle, plot the midpoints and draw the midsegments.

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

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

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

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

The graph is to the right.

Example 4: Find the slopes of \begin{align*}\overline{NM}\end{align*} and \begin{align*}\overline{QO}\end{align*}.

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

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

Example 5: Find \begin{align*}NM\end{align*} and \begin{align*}QO\end{align*}.

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

\begin{align*}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\end{align*}

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

## 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 \begin{align*}\overline{DF}\end{align*} is a midsegment of \begin{align*}\triangle ABC\end{align*}, then \begin{align*}DF = \frac{1}{2} AC = AE = EC\end{align*} and \begin{align*}\overline{DF} \| \overline{AC}\end{align*}.

Example 6a: Mark all the congruent segments on \begin{align*}\triangle ABC\end{align*} with midpoints \begin{align*}D, \ E\end{align*}, and \begin{align*}F\end{align*}.

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 \begin{align*}\triangle ABC\end{align*}, with midpoints \begin{align*}D, \ E\end{align*}, and \begin{align*}F\end{align*}.

Solution:

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

Example 7: \begin{align*}M, \ N\end{align*}, and \begin{align*}O\end{align*} are the midpoints of the sides of the triangle.

Find

a) \begin{align*}MN\end{align*}

b) \begin{align*}XY\end{align*}

c) The perimeter of \begin{align*}\triangle XYZ\end{align*}

Solution: Use the Midsegment Theorem.

a) \begin{align*}MN = OZ = 5\end{align*}

b) \begin{align*}XY = 2(ON) = 2 \cdot 4 = 8\end{align*}

c) Add up the three sides of \begin{align*}\triangle XYZ\end{align*} to find the perimeter.

\begin{align*}XY + YZ + XZ = 2 \cdot 4 + 2 \cdot 3 + 2 \cdot 5 = 8 + 6 + 10 = 24\end{align*}

Remember: No line segment over \begin{align*}MN\end{align*} means length or distance.

Example 8: Algebra Connection Find the value of \begin{align*}x\end{align*} and \begin{align*}AB\end{align*}. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are midpoints.

Solution: \begin{align*}AB = 34 \div 2 = 17\end{align*}. To find \begin{align*}x\end{align*}, set \begin{align*}3x - 1\end{align*} equal to 17.

\begin{align*}3x - 1 & = 17\\ 3x & = 18\\ x & =6\end{align*}

Know What? Revisited To the left is a picture of the \begin{align*}4^{th}\end{align*} 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.

\begin{align*}R, \ S, \ T\end{align*}, and \begin{align*}U\end{align*} are midpoints of the sides of \begin{align*}\triangle XPO\end{align*} and \begin{align*}\triangle YPO\end{align*}.

1. If \begin{align*}OP = 12\end{align*}, find \begin{align*}RS\end{align*} and \begin{align*}TU\end{align*}.
2. If \begin{align*}RS = 8\end{align*}, find \begin{align*}TU\end{align*}.
3. If \begin{align*}RS = 2x\end{align*}, and \begin{align*}OP = 20\end{align*}, find \begin{align*}x\end{align*} and \begin{align*}TU\end{align*}.
4. If \begin{align*}OP = 4x\end{align*} and \begin{align*}RS = 6x - 8\end{align*}, find \begin{align*}x\end{align*}.

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 \begin{align*}\triangle XYZ\end{align*} are 26, 38, and 42. \begin{align*}\triangle ABC\end{align*} is formed by joining the midpoints of \begin{align*}\triangle XYZ\end{align*}.
1. What are the lengths of the sides of \begin{align*}\triangle ABC\end{align*}?
2. Find the perimeter of \begin{align*}\triangle ABC\end{align*}.
3. Find the perimeter of \begin{align*}\triangle XYZ\end{align*}.
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 \begin{align*}\triangle ABC\end{align*} below find the midpoints of each side.

1. \begin{align*}A(5, -2), \ B(9, 4)\end{align*} and \begin{align*}C(-3, 8)\end{align*}
2. \begin{align*}A(-10, 1), \ B(4, 11)\end{align*} and \begin{align*}C(0, -7)\end{align*}
3. \begin{align*}A(-1, 3), \ B(5, 7)\end{align*} and \begin{align*}C(9, -5)\end{align*}
4. \begin{align*}A(-4, -15), \ B(2, -1)\end{align*} and \begin{align*}C(-20, 11)\end{align*}

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

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

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

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

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

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