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.
When an angle within a triangle is bisected, the bisector divides the triangle proportionally
By definition, divides equally, so . The proportional relationship is .
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.
Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:
Determine the value of that would make the proportion true.
You can set up this proportion just like the previous example.
Find the missing variable:
Set up a proportion and solve like in the previous examples.
Watch this video for help with the Examples above.
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.
Find the missing variables:
1. Set up a proportion and solve.
2. Set up a proportion and solve.
3. Set up a proportion and solve.
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 . What did Casey do wrong? Write the correct proportion and solve for .
Solve for the unknown variable.