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7. Congruence & Triangles
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Integrated Math 2

7.08 Midsegments of triangles

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Lesson

Concept summary

Midsegment

A midsegment of a triangle is 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.

We can justify the midsegment theorem by showing that the two triangles \triangle ADE and \triangle ABC are similar with a scale factor of 2.

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

Worked 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.

Approach

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}

Solution

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. The segment has a length of 7 x plus 3. The base of the triangle has a length of 19 x minus 9.

Approach

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.

Solution

\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

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. 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.

Approach

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.

Solution

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

Outcomes

M2.G.CO.C.8

Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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