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7.10: Proportional Triangles

  • Page ID
    5807
  • Sides divided by a line parallel to the third side of a triangle.

    Triangle Proportionality Theorem

    Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides proportionally. But what about another line that is parallel, but does not divide the other two sides into congruent halves? In fact, such a line will still divide the sides proportionally. This is called the Triangle Proportionality Theorem.

    Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

    f-d_527e8d7229a4c872dd81584e54b7d5b74ab8ea2d35c5e1705993f33b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    If \(\overline{DE}\parallel \overline{AC}\), then \(\dfrac{BD}{DA}=\dfrac{BE}{EC}\). (\(\dfrac{DA}{BD}=\dfrac{EC}{BE}\) is also a true proportion.)

    The converse of this theorem is also true.

    Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

    If \(\dfrac{BD}{DA}=\dfrac{BE}{EC}\), then \(\overline{DE}\parallel \overline{AC}\).

    What if you were given a triangle with a line segment drawn through it from one side to the other? How could you use information about the triangle's side lengths to determine if that line segment is parallel to the third side?

    Use the diagram to answers Examples 1 and 2. \(\overline{DB}\parallel \overline{FE}\).

    f-d_cf96c3c6f1d64e644976ab7ccc8c70795fa98c347abff59d021de9b8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Example \(\PageIndex{1}\)

    Name the similar triangles. Write the similarity statement.

    Solution

    \(\Delta DBC\sim \Delta FEC\)

    Example \(\PageIndex{2}\)

    \(\dfrac{FC+?}{FC}=\dfrac{?}{FE}\)

    Solution

    \(DF\); \(DB\)

    Example \(\PageIndex{3}\)

    A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?

    f-d_6a68693d149bd1df7051f07cc3b277885d78e3f07f2e7e0ea1f6b4e1+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Solution

    The midsegment splits the sides evenly. The ratio would be 8:8 or 10:10, which both reduce to 1:1.

    Example \(\PageIndex{4}\)

    In the diagram below, \(\overline{EB}\parallel \overline{CD}\). Find \(BC\).

    f-d_13e4cbbf6ccf4d9c24fd0ec217c40649ba2c89c6d1207427128d93c0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    To solve, set up a proportion.

    \(\begin{aligned} \dfrac{10}{15}=\dfrac{BC}{12} \rightarrow 15(BC)&=120 \\ BC&=8 \end{aligned}\)

    Example \(\PageIndex{5}\)

    Is \(\overline{DE}\parallel \overline{CB}\)?

    f-d_9a9cae937c4a24316973e566ba77a0cf95e3a5aedc18bbc7c3fdd076+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    If the ratios are equal, then the lines are parallel.

    \(\dfrac{6}{18}=\dfrac{8}{24}=\dfrac{1}{3}\)

    Because the ratios are equal, \(\overline{DE}\parallel \overline{CB}\).

    Review

    Use the diagram to answer questions 1-7. \(\overline{AB}\parallel \overline{DE}\).

    f-d_42be0dd70828a51e6167efdead147579fb8347a98f23552861dc93f9+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)
    1. Find \(BD\).
    2. Find \(DC\).
    3. Find \(DE\).
    4. Find \(AC\).
    5. What is \(BD:DC\)?
    6. What is \(DC:BC\)?
    7. Why \(BD:DC\neq DC:BC\)?

    Use the given lengths to determine if \(\overline{AB}\parallel \overline{DE}\).

    1. f-d_59d0dfc411a8d2f59a82b98742a7bd680d3b562482257247dfb6e8df+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. f-d_bb48652d23190094982b04039eadde91e7e5436b8aafc98fd0bbe39d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    3. f-d_0101d35ec6a09a5a3e01aa4ed7b2d258485413dcca1fe2b94262d7c9+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    4. f-d_f6cb319fd000335345ca3bf2eacb0b2b057fea534e9344d1529e663c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    5. f-d_ccc46709a6f79de6610e1420eef68924e89f88f52e8ed49c2884d0b2+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    6. f-d_5c775b6ca60823e2042d63b35f1d2ac306fc6517c0cdceb0ecfaf493+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 7.8.

    Resources

    Vocabulary

    Term Definition
    Congruent Congruent figures are identical in size, shape and measure.
    midsegment A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.
    Parallel Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.
    Proportion A proportion is an equation that shows two equivalent ratios.
    Triangle Proportionality Theorem The Triangle Proportionality Theorem states that if a line is parallel to one side of a triangle and it intersects the other two sides, then it divides those sides proportionally.
    Triangle Proportionality Theorem Converse The Triangle Proportionality Theorem converse states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

    Additional Resources

    Video: Using the Properties of the Triangle Proportionality Theorem to Solve for Unknown Values

    Activities: Triangle Proportionality Discussion Questions

    Study Aids: Proportionality Relationships Study Guide

    Practice: Proportional Triangles

    Real World: Triangle Proportionality

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