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? After completing this Concept, you'll be able to solve such problems.
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CK-12 Foundation: Chapter7ProportionswithAngleBisectorsA
James Sousa: Triangle Angle Bisector Theorem
James Sousa: Using the Triangle Angle Bisector Theorem to Determine Unknown Values
Guidance
When an angle within a triangle is bisected, the bisector divides the triangle proportionally
By definition, \begin{align*}\overrightarrow{AC}\end{align*} divides \begin{align*}\angle BAD\end{align*} equally, so \begin{align*}\angle BAC \cong \angle CAD\end{align*}. The proportional relationship is \begin{align*}\frac{BC}{CD}=\frac{AB}{AD}\end{align*}.
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.
Example A
Find \begin{align*}x\end{align*}.
Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:
\begin{align*}\frac{9}{x} &= \frac{21}{14}\\ 21x &= 126\\ x &= 6\end{align*}
Example B
Determine the value of \begin{align*}x\end{align*} that would make the proportion true.
You can set up this proportion just like the previous example.
\begin{align*}\frac{5}{3} &= \frac{4x+1}{15}\\ 75 &= 3(4x+1)\\ 75 &= 12x+3\\ 72 &= 12x\\ 6 &= x\end{align*}
Example C
Find the missing variable:
Set up a proportion and solve like in the previous examples.
\begin{align*}\frac{12}{4}&=\frac{x}{3}\\ 36&=4x\\ x&=9\end{align*}
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7ProportionswithAngleBisectorsB
Vocabulary
Pairs of numbers are proportional if they are in the same ratio. An angle bisector is a ray that divides an angle into two congruent angles.
Guided Practice
Find the missing variables:
1.
2.
3.
Answers:
1. Set up a proportion and solve.
\begin{align*} \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10 \end{align*}
2. Set up a proportion and solve.
\begin{align*} \frac{20}{y}&=\frac{15}{28-y}\\ 15y&=20(28-y)\\ 15y&=560-20y\\ 35y&=560\\ y&=16\end{align*}
3. Set up a proportion and solve.
\begin{align*} \frac{12}{z}&=\frac{15}{9-z}\\ 15z&=12(9-z)\\ 15z&=108=12z\\ 27z&=108\\ z&=4\end{align*}
Practice
Find the value of the missing variable(s).
Find the value of each variable in the pictures below.
Find the unknown lengths.
- Error Analysis
Casey attempts to solve for a in the diagram using the proportion \begin{align*}\frac{5}{a}=\frac{6}{5}\end{align*}. What did Casey do wrong? Write the correct proportion and solve for \begin{align*}a\end{align*}.
Solve for the unknown variable.