In this explainer, we will learn how to use the theorem of angle bisector and its converse to find a missing side length in a triangle.
Let’s consider a triangle and the angle bisector of angle , .
Angle Bisector Theorem (Part I)
If an interior angle of a triangle is bisected, the bisector divides the opposite side into segments whose lengths are in the same ratio as the lengths of the other sides of the triangle.
How to Prove the Angle Bisector Theorem (Part I)
is parallel to . We have
Therefore, , which means that is an isosceles triangle, and so .
As and are parallel, the triangles and are similar. It follows that which can be rewritten as
By multiplying both sides of the equation by , we get
Expanding the brackets, we find
Taking away , which is the same as , from both sides, we get
Dividing now both sides by , we have
And since , , and , we finally have
Let’s see with an example how to apply this theorem.
Example 1: Applying the Angle Bisector Theorems
In the figure, bisects , , , and the perimeter of is 57. Determine the lengths of and .
bisects in triangle . From the angle bisector theorem (theorem I), we know that we have
And we know as well that
We know that and , so
By substituting this into our first equation, we get
Let’s look at another theorem, still in triangle whose angle bisector of angle is . This theorem is about the length of .
Angle Bisector Theorem (Part II)
In any triangle , if is the angle bisector of angle , then we have
Let’s see how to use this theorem.
Example 2: Applying the Angle Bisector Theorems
In the triangle , , , and . Given that bisects and intersects at , determine the length of .
bisects in triangle . From the angle bisector theorem (Part II), we know that we have
From the question, we know the lengths of , , and . Therefore, to find , we must first find . For this, we apply the angle bisector theorem (Part I):
Now, we can plug in all the lengths:
There is a third angle bisector theorem, which concerns the bisector of an exterior angle of a triangle.
Angle Bisector Theorem (Part III)
The bisector of an exterior angle of a triangle intersects an extension of the opposite side of the triangle at . The ratio of the length of only the extension of the opposite side to the length of the opposite side plus the extension is equal to the ratio of the lengths of the other two sides of the triangle:
Let’s see an example on how to use this theorem.
Example 3: Applying the Angle Bisector Theorems
Given that , , and , what is ?
is the angle bisector of the exterior angle of . Therefore, we have
Since , we can write
And by plugging in the lengths given in the question,
By multiplying both sides by , we get