7.13: Proportions and Angle Bisectors
- Page ID
- 5918
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
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:
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:
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 \(x\).
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.
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:
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).
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Solve for the unknown variable.
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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