Topics
3. Triangles
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United States of America
Grades 9-12

3.06 Midsegments

Lesson
Practice

Introduction

We will be introduced to midsegments, extending the concept of segment bisectors to get a new type of triangle construction, and explore the relationships that arise from this construction. We will focus on identifying and constructing midsegments, then using the midsegment theorem to determine measures and thus solve problems.

Midsegments

Exploration

Drag the points to change the triangle. Check the box to show the segment lengths.

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  1. What do you notice about the segments that are the same color?
Midsegment

A line segment that joins the midpoints of two sides of the triangle

A triangle with a dashed line segment starting at the midpoint of one side and ending at the midpoint of an adjacent side.
Triangle A B C. Point D is the midpoint of A B, and point E is the midpoint of A C. A segment is drawn from D to E. Segment D E is parallel to B C.

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

In this example, \overline{DE} is the midsegment connecting the midpoints of \overline{AB} and \overline{AC}.

This then means that \overline{DE} \parallel \overline{BC} and DE=\dfrac{1}{2}BC.

Triangle B C D with midpoint G on B C, midpoint F on C D, and midpoint E on B D. Segments G F, F E, and G E are drawn.

There are three midsegments in any triangle, connecting each possible pair of midpoints.

We can verify through experimentation that the midsegment theorem is true. We will formally prove that it is true in a future lesson.

Examples

Example 1

Identify the midsegment(s) of the given triangle.

Triangle A B C. Point D is the midpoint of A B, point F is the midpoint of A C, and point E is the midpoint of B C. Segments are drawn from D to E, from A to E, from E to F, and from B to F.
Worked Solution
Create a strategy

A midsegment joins the midpoints of two sides. We are looking for line segments that do this.

There are four lines segments that are not sides of the triangle.

  • \overline{DE}
  • \overline{EF}
  • \overline{EA}
  • \overline{BF}
Apply the idea

We can now go through these line segments to see which ones fit the definition of a midsegment and which ones do not.

  • \overline{DE} is a midsegment as both D and E are midpoints of sides of the triangle.
  • \overline{EF} is a midsegment as both E and F are midpoints of sides of the triangle.
  • \overline{EA} is not a midsegment as one the endpoints is a vertex, not a midpoint.
  • \overline{BF} is not a midsegment as one the endpoints is a vertex, not a midpoint.

Example 2

Find the value of x.

A triangle. A segment is drawn inside the triangle, connecting the midpoints of the two sides of the triangle. One side of the triangle makes a 120 degree angle with the segment. The same side of the triangle makes an x degree angle with the base of the triangle.
Worked Solution
Create a strategy

The line segment has endpoints on the midpoints of two sides of the triangle, so we can say it is a midsegment. We can then use the midsegment theorem to show that the segment connecting the two sides is parallel to the third side. We can then use the fact that the two labelled angles are corresponding angles.

Apply the idea

Since corresponding angles on parallel lines are congruent, we must have that x=120.

Example 3

Find the value of x.

A triangle. A segment is drawn inside the triangle, connecting the midpoints of the two sides of the triangle. The segment has a length of 7 x plus 3. The base of the triangle has a length of 19 x minus 9.
Worked Solution
Create a strategy

We can see that the line segment of length 7x+3 is a midsegment. We can use the midsegment theorem to show that this line segment will be half as long as the line segment of length 19x-9. We can then relate the two lengths using an equation and solve for x.

Apply the idea
\displaystyle 7x+3\displaystyle =\displaystyle \frac{1}{2}(19x-9)Midsegment theorem
\displaystyle 14x+6\displaystyle =\displaystyle 19x-9Multiply both sides by 2
\displaystyle 15\displaystyle =\displaystyle 5xSubtract 14x and add 9 to both sides
\displaystyle x\displaystyle =\displaystyle 3Divide both sides by 5
Reflect and check

Confirm that the midsegment is half the length of the base of the triangle by substituting x in the original expressions.

\displaystyle 7x+3\displaystyle =\displaystyle 7(3)+3Substitute x=3
\displaystyle =\displaystyle 24Evaluate the multiplication and addition
\displaystyle 19x-9\displaystyle =\displaystyle 19(3)-9Substitute x=3
\displaystyle =\displaystyle 48Evaluate the multiplication and addition

The length of the base of the triangle is 48 units and the length of the midsegment is 24 units.

Example 4

Construct a midsegment in a triangle.

Worked Solution
Create a strategy

Consider the triangle shown:

Triangle D E F.

We will use a compass and straightedge to construct the midpoint of \overline{DE} followed by a line through the midpoint parallel to \overline{DF}.

Apply the idea

  1. Identify the segment we want to bisect, then open the compass width to just past half the segment's length and draw an arc from one endpoint that extends to both sides of the segment.

    A diagram showing the first step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  2. Without changing the compass width, draw another arc from the other endpoint that intersects the original arc on both sides of the segment.

    A diagram showing the second step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  3. Label the intersection of the arcs with points and connect them.

    A diagram showing the third step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  4. The intersection of the line with the segment is the midpoint of \overline{DE}.

    A diagram showing the fourth step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  5. To construct the line parallel to \overline{DF} through M, set the compass width to the distance MF.

    A diagram showing the fifth step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  6. Construct an arc centered at D with the radius MF.

    A diagram showing the sixth step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  7. Set the compass width to the distance MD.

    A diagram showing the seventh step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  8. Construct an arc centered at F with the radius MD.

    A diagram showing the eighth step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.

  9. Construct a line through M and the intersection of the two arcs. This line is the midsegment of \triangle{DEF}.

    A diagram showing the final step in constructing the midsegment of triangle D E F. Speak to your teacher for more details.
Reflect and check

To check that we've accurately drawn a midsegment, use a ruler to measure the length of the base of the triangle drawn and the length of the midsegment drawn. The length of the midsegment should be half the length of the base.

Idea summary

A midsegment of a triangle is a line segment that joins the midpoints of two sides of the triangle. Its length is equal to half the length of the parallel side of the triangle.

Outcomes

G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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