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7.10: Proportions with Angle Bisectors

Difficulty Level: At Grade Created by: CK-12
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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, AC divides BAD equally, so BACCAD. The proportional relationship is BCCD=ABAD.

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 x.

Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:

9x21xx=2114=126=6

Example B

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

You can set up this proportion just like the previous example.

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

Example C

Find the missing variable:

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

12436x=x3=4x=9

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.

20820yy=25y=200=10

2. Set up a proportion and solve.

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

3. Set up a proportion and solve.

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

Practice

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 5a=65. What did Casey do wrong? Write the correct proportion and solve for a.

Solve for the unknown variable.

Notes/Highlights Having trouble? Report an issue.

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Vocabulary

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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Difficulty Level:
At Grade
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Date Created:
Jul 17, 2012
Last Modified:
Aug 02, 2016
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