Lesson On what segments the angle bisector divides the side of a triangle

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On what segments the angle bisector divides the side of a triangle

Theorem
In a triangle, the angle bisector divides the side to which it is drawn, in two segments proportional to the ratio of two other sides of a triangle. Prove.

Proof
We have a triangle ABC (Figure 1) with the vertexes
A, B and C. The sides of the triangle BC and AC are of
the length a and b respectively, as shown in the Figure 1.
The angle bisector CD of the angle C divides the side AB
into segments AD and BD of the length y and x respectively.
The Theorem states that x%2Fy+=+a%2Fb

.

We will use the Law of sines. It was established in the
lessons Law of sines and Law of sines - the Geometric
Proof that are under the topic Triangles of the section
Geometry in this site.

Figure 1. To the Theorem 1

Figure 2. To the proof of the Theorem

So, let us draw perpendiculars DE and DF from the point D to the sides AC and BC of the triangle till the intersection points E and F respectively (Figure 2).
Since CD is the angle bisector, the point D is equidistant from the sides AC and BC of the angle ACB (see the lesson An angle bisector properties under the topic Triangles of the section Geometry in this site). Therefore, the perpendiculars DE and DF are of equal length. If we denote this length as z, then (Figure 2)

sin%28A%29=z%2Fy

.

Now apply the Law of sines for the triangle ABC in the form

a%2Fsin%28A%29+=+b%2Fsin%28B%29

and substitute the expressions for sin%28A%29

and

from the above. After cancellation and transformations you will get

x%2Fy+=+a%2Fb

,

which is exactly what have to be demonstrated. The proof is completed.

Below are examples that show how to apply the proved Theorem.

Problem 1
The triangle has the sides of 4 cm, 5 cm and 6 cm long.
Find the length of the segments the angle bisector cuts the side of 5 cm.

Solution
Let

and

be the length of the segments the angle bisector cuts the side of 5 cm.
Then you have the system of two equations for

and

system%28x+%2B+y+=+5%2C%0D%0Ax%2Fy+=+4%2F6%29

The latest equation is based on the Theorem above.

To solve the system, express

from the second equation

x+=+4%2F6%2Ay+=+2%2F3%2Ay

and then substitute it to the first equation. You get

2%2F3%2Ay+%2B+y+=+5

and then

.

Answer. The segments the angle bisector cuts the side of 5 cm are 2 cm and 3 cm long.

Problem 2
The triangle has the perimeter of 60 cm. (60 = 4*(4 + 5 +6) = 16 + 20 + 24)
The angle bisector cuts one side of the triangle into segments of 8 and 12 cm.
Find two other sides of the triangle.

Solution
Let

and

be the length of two other sides of the triangle.
The length of the first side of the triangle is 8 cm + 12 cm = 20 cm.
Hence, the summary length of two other sides of the triangle is equal to 60 cm - 20 cm = 40 cm.
Therefore, you have the system of two equations

and

system%28x+%2B+y+=+40%2C%0D%0Ax%2Fy+=+8%2F12%29

The latest equation is based on the Theorem above.

To solve the system, express

from the second equation

x+=+8%2F12%2Ay+=+2%2F3%2Ay

and then substitute it to the first equation. You get

2%2F3%2Ay+%2B+y+=+40

and then

.

Answer. Two other sides of the triangle are 16 cm and 24 cm long.

Appendix
Having the Theorem proved, we can easily deduce the formulas
expressing the length of segments to which the angle bisector
cuts the side of the triangle.

Indeed, let

and

be the length of the segments the angle
bisector drawn between the sides of triangle

and

cuts the side

.
Then you have the system of two equations for

and

system%28x+%2B+y+=+c%2C%0D%0Ax%2Fy+=+a%2Fb%29

Figure 1. To the Appendix

The latest equation is based on the Theorem above.

To solve the system, express

from the second equation

x+=+a%2Fb%2Ay

and then substitute it to the first equation. We get

a%2Fb%2Ay+%2B+y+=+c

(after multiplication both sides of the previous equation by

.

Now substitute this expression for

to the formula above for

. We get

x+=+%28a%2Fb%29%2A%28%28b%2Ac%29%2F%28a+%2B+b%29%29+=+%28a%2Ac%29%2F%28a+%2B+b%29

.

Thus the final result is: The angle bisector drawn between sides and of a triangle cuts the side into segments of the length of and .

For navigation over the lessons on Properties of Triangles use this file/link Properties of Trianles.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link GEOMETRY - YOUR ONLINE TEXTBOOK.

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