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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 BACCAD, then BCCD=ABAD.

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?

 

 

Examples

Example 1

Fill in the missing variable:

Set up a proportion and solve.

20y15y15y35yy=1528y=20(28y)=56020y=560=16

Example 2

Fill in the missing variable:

Set up a proportion and solve.

12z15z15z27zz=159z=12(9z)=108=12z=108=4

Example 3

Find x.

The ray is the angle bisector and it splits the opposite side in the same ratio as the other two sides. The proportion is:

9x21xx=2114=126=6

Example 4

Find the value of x that would make the proportion true.

You can set up this proportion like the previous example.

537575726=4x+115=3(4x+1)=12x+3=12x=x

Example 5

Find the missing variable:

Set up a proportion and solve like in the previous examples.

12436x=x3=4x=9

Review

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. 

Resources

 

Vocabulary

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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