Lesson 5

Splitting Triangle Sides with Dilation, Part 1

  • Let’s draw segments connecting midpoints of the sides of triangles.

5.1: Notice and Wonder: Midpoints

Here’s a triangle \(ABC\) with midpoints \(L, M\), and \(N\).

Triangle A B C with midpoints M, L and N drawn forming a triangle.

What do you notice? What do you wonder?

5.2: Dilation or Violation?

Here’s a triangle \(ABC\). Points \(M\) and \(N\) are the midpoints of 2 sides.

Triangle A B C. Point M is midpoint of A B. Point N is midpoint of A C. Segment M N is drawn.  On both segments A M and M B, one tick mark. On both segments A N and C N, two tick marks.   
  1. Convince yourself triangle \(ABC\) is a dilation of triangle \(AMN\). What is the center of the dilation? What is the scale factor?
  2. Convince your partner that triangle \(ABC\) is a dilation of triangle \(AMN\), with the center and scale factor you found.
  3. With your partner, check the definition of dilation on your reference chart and make sure both of you could convince a skeptic that \(ABC\) definitely fits the definition of dilation.
  4. Convince your partner that segment \(BC\) is twice as long as segment \(MN\).
  5. Prove that \(BC=2MN\). Convince a skeptic.

5.3: A Little Bit Farther Now

Here’s a triangle \(ABC\). \(M\) is \(\frac23\) of the way from \(A\) to \(B\). \(N\) is \(\frac23\) of the way from \(A\) to \(C\).

Triangle A B C with segment M N drawn from A B to A C.

What can you say about segment \(MN\), compared to segment \(BC\)? Provide a reason for each of your conjectures.



  1. Dilate triangle \(DEF\) using a scale factor of -1 and center \(F\).
  2. How does \(DF\) compare to \(D'F'\)?
  3. Are \(E\), \(F\), and \(E'\) collinear? Explain or show your reasoning.
Triangle D E F.

Summary

Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If \(D\) is the midpoint of segment \(BC\) and \(E\) is the midpoint of segment \(BA\), then what can we say about \(ED\) and triangle \(ABC\)?

Segment \(ED\) is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if \(AC=10\), then \(ED=5\). This happens because the entire triangle \(EBD\) is a dilation of triangle \(ABC\) with a scale factor of \(\frac12\).

\(\overline{BD}\cong\overline{DC},\overline{BE}\cong\overline{EA}\)

Triangle A B C. Point E is midpoint of segment B A. Point D is midpoint of segment B C. Segment E D drawn. On both segments A E and E B, two tick marks. On both segments B D and C D, one tick mark.  

In triangle \(ABC\), segment \(FG\) divides segments \(AB\) and \(CB\) proportionally. In other words, \(\frac{BG}{GA}\)=\(\frac{BF}{FC}\). Again, there is a dilation that takes triangle \(ABC\) to triangle \(GBF\), so \(FG\) is parallel to \(AC\) and we can calculate its length using the same scale factor.

\(\overleftrightarrow{FG} \parallel \overleftrightarrow{AC}\)

Triangle A B C. Point G on segment A B. Point F on segment B C. Directed line segments A C and G F drawn with arrows point towards Points C and F.  

Glossary Entries

  • dilation

    A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.

    Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).

  • scale factor

    The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.