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Proportions with Angle Bisectors

Angle bisectors divide triangles proportionally.

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Proportions with Angle Bisectors

Proportions with Angle Bisectors 

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.

Solving for Unknown Values 

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*}

Solving for an Unknown Value that will make a Proportion True 

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*}

Finding a Missing Variable 

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*}


Find the missing variables:

Example 1

Set up a proportion and solve.

\begin{align*} \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10 \end{align*}

Example 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*}

Example 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*}


Find the value of the missing variable(s).

Find the value of each variable in the pictures below.

Find the unknown lengths.

  1. 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.

Review (Answers)

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

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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”.

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