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Proportions with Angle Bisectors

Angle bisectors divide triangles proportionally.

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Proportions with Angle Bisectors

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 \begin{align*}\triangle BAC \cong \triangle CAD\end{align*}, then \begin{align*}\frac{BC}{CD} = \frac{AB}{AD}\end{align*}.

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 1

Fill in the missing variable:

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 2

Fill in the missing variable:

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*}

Example 3

Find \begin{align*}x\end{align*}.

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{align*}\frac{9}{x} &= \frac{21}{14}\\ 21x &= 126\\ x &= 6\end{align*}

Example 4

Find the value of \begin{align*}x\end{align*} that would make the proportion true.

You can set up this proportion 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 5

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 value of the missing variable(s).

Solve for the unknown variable.

Review (Answers)

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


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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”.

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