7.13: Proportions and Angle Bisectors
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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