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# Angle Bisector – Definition, Construction, Properties

## What Is an Angle Bisector?

An angle bisector is a line or ray that divides an angle into two equal parts. The angle bisector passes through the vertex of the angle and divides it into two smaller angles, each of which has the same measure. The angle bisector is sometimes also called the angle tri-sector because it divides the angle into three equal parts.

According to the angle bisector theorem, the ratio of the length of the side opposite the angle to the length of the other side is equal to the ratio of the other two sides of the triangle.

For example, consider a triangle ABC, where angle A is bisected by the angle bisector AD. Then, according to the angle bisector theorem, the following equation holds true:

AB/AC = BD/DC

Here, AB and AC are the lengths of the two sides of the triangle that form the angle at vertex A, and BD and DC are the lengths of the two segments into which the angle bisector AD divides the side BC.

## Properties of Angle Bisector

Here are given some of the main properties of the angle bisector:

1. It divides an angle into two congruent angles
2. Angle Bisector intersects the opposite side at a point that is equidistant from the other two sides
3. It can be used to find the location of the incenter of a triangle
4. It can be used to find the angle between two lines
5. It can be used to find the focal point of a lens in optics

## How to Construct an Angle Bisector?

Constructing an angle bisector involves drawing a line that splits an angle into two congruent angles. Here are the steps to construct an angle bisector:

Step 1: Draw the given angle

Start by drawing the given angle using a straightedge and a pencil. Label the vertex of the angle as point A and the two sides of the angle as AB and AC.

Step 2: Place the compass on point A

Place the pointed end of the compass on point A and adjust the compass to a length that is greater than half the length of the angle.

Step 3: Draw an arc that intersects AB and AC

Using the compass, draw an arc that intersects side AB and side AC of the angle. Label the points of intersection as points D and E, respectively.

Step 4: Draw a line through point A and point D

Using the straightedge, draw a line that passes through point A and point D.

Step 5: Draw a line through point A and point E

Using the straightedge, draw a line that passes through point A and point E.

Step 6: The intersection of the two lines is the angle bisector

The point where the two lines intersect is the angle bisector of the angle. Label this point as point F.

Now, the line AF is the angle bisector of angle BAC, which divides it into two congruent angles, angle BAF and angle CAF.

### Solved Examples on Angle Bisector

Here are some solved examples that demonstrate the use of angle bisectors in geometry:

##### Example 1: In the triangle ABC, angle A is bisected by the line segment AD, and AB = 8 cm, AC = 10 cm, and BC = 12 cm. Find the length of AD.

Solution:
Let’s use the angle bisector theorem to find the length of AD. According to the theorem:

AB/BD = AC/CD

Substituting the given values, we get:

8/BD = 10/CD

BD/CD = 8/10 = 4/5

We know that the sum of the lengths of BD and CD is equal to the length of BC, so:

BD + CD = 12

Substituting BD/CD = 4/5, we get:

BD = (4/9) x 12 = 16/3 cm

CD = (5/9) x 12 = 20/3 cm

Therefore, the length of AD is:

AD = BD + CD = 16/3 + 20/3 = 36/3 = 12 cm

##### Example 2: Find the angle between the lines y = 2x + 3 and y = -3x + 7.

Solution:
Let’s find the angle bisector of the angle between the two lines. The angle bisector will be a line that divides the angle between the two lines into two congruent angles.

The slopes of the two lines are 2 and -3, respectively. The angle between the lines can be found using the formula:

tan θ = (m2 – m1)/(1 + m1m2)

where m1 and m2 are the slopes of the two lines.

Substituting the values, we get:

tan θ = (-3 – 2)/(1 – 6) = 5/5 = 1

θ = tan^-1(1) = 45 degrees

Therefore, the angle between the lines is 45 degrees. Since the angle bisector divides the angle into two congruent angles, each of the angles between the angle bisector and the two lines will be 22.5 degrees.

##### Example 3: In triangle ABC, angle A is bisected by the line segment AD, and the length of AD is 6 cm. If AB = 8 cm and AC = 10 cm, find the length of BC.

Solution:
Let’s use the angle bisector theorem to find the length of BC. According to the theorem:

AB/BD = AC/CD

Substituting the given values, we get:

8/BD = 10/CD

BD/CD = 8/10 = 4/5

We also know that AD bisects angle A, so angle BAD and angle CAD are congruent. This means that the triangle ABD is similar to the triangle ACD.

Using similar triangles, we can set up the following equation:

Substituting the values we know, we get:

4/5 = CD/(CD + 6)

Solving for CD, we get:

CD = 30/7 cm

Therefore, the length of BC is:

BC = BD + CD = (4/5) x 12 + 30/7 = 102/35 cm

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