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7.13: Proportions and Angle Bisectors

  • Page ID
    5918
  • Angle bisectors divide triangles proportionally.

    Angle Bisector Theorem

    When an angle within a triangle is bisected, the bisector divides the triangle proportionally. This idea is called the Angle Bisector Theorem.

    Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

    f-d_da84f356665739ba245a4ebb575e160d0958b5bf84667475b08ed765+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    If \(\Delta BAC\cong \Delta CAD\), then \(\dfrac{BC}{CD}=\dfrac{AB}{AD}\).

    What if you were told that a ray was an angle bisector of a triangle? How would you use this fact to find unknown values regarding the triangle's side lengths?

    Example \(\PageIndex{1}\)

    Fill in the missing variable:

    f-d_59445c1a5ee6fb90b94191437dc40e381f1301e8c0f36373bb62a003+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Solution

    Set up a proportion and solve.

    \(\begin{aligned} \dfrac{20}{y}&=\dfrac{15}{28−y} \\ 15y&=20(28−y) \\ 15y&=560−20y \\ 35y&=560 \\ y&=16\end{aligned}\)

    Example \(\PageIndex{2}\)

    Fill in the missing variable:

    f-d_557b73cc3160dbb5c81c50c78be710a47dc53e0960bdbb66a795bf41+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Solution

    Set up a proportion and solve.

    \(\begin{aligned}\dfrac{12}{z}&=\dfrac{15}{9−z} \\ 15z&=12(9−z) \\ 15z&=108-12z \\ 27z&=108 \\ z&=4\end{aligned}\)

    Example \(\PageIndex{3}\)

    Find \(x\).

    f-d_077dc3197010de17cdc9357863a8fa7207bcb1674d7407b30163ae1a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    The ray is the angle bisector and it splits the opposite side in the same ratio as the other two sides. The proportion is:

    \(\begin{aligned} \dfrac{9}{x}&=\dfrac{21}{14} \\ 21x&=126 \\ x&=6\end{aligned}\)

    Example \(\PageIndex{4}\)

    Find the value of \(x\) that would make the proportion true.

    f-d_bbb063d9126edadad133fa633864c5ee76c661b4186d65737784c78e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    You can set up this proportion like the previous example.

    \(\begin{aligned} \dfrac{5}{3}&=\dfrac{4x+1}{15} \\ 75&=3(4x+1) \\ 75&=12x+3 \\ 72&=12x \\ 6&=x\end{aligned}\)

    Example \(\PageIndex{5}\)

    Find the missing variable:

    f-d_2d44b37c3732899c8d3b2f2255f038ab9e6128795420da3587c8b3fe+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    Set up a proportion and solve like in the previous examples.

    \(\begin{aligned}\dfrac{12}{4}&=\dfrac{x}{3} \\ 36&=4x \\ x&=9\end{aligned}\)

    Review

    Find the value of the missing variable(s).

    1. f-d_24ce554d011a0e3e31e7d4daf635b98a120cb9552ea8ca3d096f76d3+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. f-d_84d6db3c4d7396df4b25083fc71a31b4099423bcf9a2f4c42f481080+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)

    Solve for the unknown variable.

    1. f-d_95564529aef0e07340e7594459e714692c4c70a0311fccc8d5cf3c01+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    2. f-d_c53bf9b7124a0c8e3c882cd0e8fdc89598c464a367cc28423c588895+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    3. f-d_d02c7214bd6fb963ee4df4fd635bac41dd929565555dfbbedc3c497f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    4. f-d_4f6cd7437b63ded507ee66f64de8804a1c5affe643490a39d1b42551+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    5. f-d_b15d430eab396d9b58d837d2fbab5eaecd7a33e5cb6b2c69ae0cbd35+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)
    6. f-d_19860cb0860db486a4d0f2f59517212d4bc3a1bd01c0e3f194eff5df+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    7. f-d_744d19ef65fa873e8cf1e31568507ae5ea6479dc3f2a688982490511+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    8. f-d_ac393562e46f317a0ef6d90fb60756782e3a440c6dd1e286d08688e8+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)
    9. f-d_149663e96160a305980f5d9e3fc57f54677b641e80af1468a2c22b65+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    10. f-d_0d13492eeffe41cc44956ed431e5bac831e460fdc0a942f858da9511+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    11. f-d_8b4b09a8ece9e319e976f730c5630b5fa008d09c83c24fd3022621f6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    12. f-d_af89d4ac2707cffde585835c904f60b72afcec506d078ae190534e82+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{20}\)
    13. f-d_9e9ddd708fa1553836a236280ab24fe8e88db325de3c424019d9e5f8+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 7.10.

    Resources

    Vocabulary

    Term Definition
    angle bisector A ray that divides an angle into two congruent angles.
    Angle Bisector Theorem The angle bisector theorem states that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
    Proportion A proportion is an equation that shows two equivalent ratios.
    Ratio A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.

    Additional Resources

    Interactive Element

    Video: Using the Properties of the Triangle Angle Bisector Theorem to Determine Unknown Values

    Activities: Proportions with Angle Bisectors Discussion Questions

    Study Aids: Proportionality Relationships Study Guide

    Real World: Triangle Proportionality