**Bisector** is a line that divides a line or an angle in to two equivalent parts. There are two types of Bisectors based on what geometrical shape it bisects.

- Bisector of a Line
- Angle Bisector

Bisector of a line divides a line in to two equal parts. Angle bisector divides the angle in to two angles of equal measures.

A bisector is a straight line that divides any figure into two equal parts. An angle or a line is bisected when they are cut into 2 equal parts.
### Line Segment Bisector

An** angle bisector ** divides an angle into equal
angles. If the angle is x$^o$, each of the two angles formed will now be
$(\frac{x}{2})^o$. The angle bisector passes through the vertex of the angle.

The figure below shows an angle bisector of $\angle$ ABC.

### Angle Bisector Theorem

The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle.

### Angle Bisector Conjecture

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

### Angle Bisector Problems

Find the value of x, if $\bar{BD}$ is the angle bisector of $\angle$ABC.

Since $\bar{BD}$ is the angle bisector of $\angle$ABC

=> 2x = 80$^o$

x = 40$^o$ (Divide each side by 2)

Thus, the value of x is 40$^o$. → Read More An angle bisector of a triangle is a bisector of an angle of the triangle. An angular bisector of a triangle is the line segment which bisect an angle of the triangle and has its other endpoint on the opposite side of that angle.

### Triangle Bisector Theorem

An angle bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.

In this figure, $\frac{AB}{AC} = \frac{BD}{DC}$

A perpendicular bisector is a segment, ray, or line that intersects a given segment at a 90$^o$, and passes through its mid point.

### Construct Perpendicular Bisector

To construct the perpendicular bisector of a segment follow the following steps:

**Step 1: **Draw a line segment $\bar{PQ}$.

**Step 2: **Set compass to more than half the distance between P and Q. Using P as center, swing an arc one side of the segment.

Step 3: With same setting of the compass, swing another arc intersecting the first, using Q as a center. Label the intersecting point as R.

Step 4: The point where the two arcs intersect is equidistant from the endpoints of PQ. Join R and the mid point of PQ, named as M.

### Perpendicular Bisector Theorem

A point lies on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.

In this figure: AH = BH, AG = BG, AF = BF and AE = BE.

### Perpendicular Bisector Equation

The equation of the perpendicular bisector of the segment joining the points ($x_1, y_1$) and ($x_2, y_2$).

Equation of $\bar{RM}$ is:

y - $\frac{y_1 + y _2}{2}$ = - $\frac{x_2 - x_1}{y_2 - y _1}$ (x - $\frac{x_1 + x _2}{2}$)

The slope of perpendicular bisector = Negative reciprocal of line segment's slope.

### Perpendicular Bisector Triangle

The perpendicular bisectors of the sides of a triangle intersect at a point. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.

### Perpendicular bisector conjecture

If a point is on the bisector of an line, then it is equidistant from the endpoints of the segment.

### Find Perpendicular Bisector

Find the equation of the perpendicular bisector of the line segment joining the points (1, 2) and (3, 6).

Step 1: Find the slope and mid point of the line segment.

Slope of the line segment joining the points (1, 2) and (3, 6)

Slope (m) = $\frac{6-2}{3-1}$ = $\frac{4}{2}$ = 2

Mid point: ($\frac{1+3}{2}$, $\frac{2+6}{2}$) = (2, 4)

Step 2:Slope of perpendicular bisector, $m_1$ = $\frac{-1}{m}$ = $\frac{-1}{2}$

Step 3: Equation of the bisector passing through mid point of line segment is

y - 4 = $\frac{-1}{2}$(x - 2)

2y - 8 = -x + 2

2y = -x + 10

or y = -0.5x + 5, which is required equation.

### Perpendicular Bisector Problems

Given below are some of the practice problems on perpendicular bisector.

### Practice Problems

**Question 1: **Construct a perpendicular bisector for $\bar{AB}$. If a line segment of length 12 cm.

**Question 2: **Find the equation of the perpendicular bisector of the line segment joining the points (4, 9) and (10, 3).

Given below are the examples on bisector. ### Solved Examples

**Question 1: **Find the missing side length

$\bar{CA} = 46, \bar{CD} = 49, \bar{BA} = 20, \bar{DB} = ?$

** Solution: **
**Question 2: **

** Solution: **

A bisector divides a figure into two congruent parts. The bisector of a segment always contains the midpoint of the segment.

A **line bisector** divides the line segment into 2 equal parts. It passes through the midpoint of the line segment. In the below figure line PQ is the bisector of AB.

An

The figure below shows an angle bisector of $\angle$ ABC.

The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle.

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Find the value of x, if $\bar{BD}$ is the angle bisector of $\angle$ABC.

Since $\bar{BD}$ is the angle bisector of $\angle$ABC

=> 2x = 80$^o$

x = 40$^o$ (Divide each side by 2)

Thus, the value of x is 40$^o$. → Read More An angle bisector of a triangle is a bisector of an angle of the triangle. An angular bisector of a triangle is the line segment which bisect an angle of the triangle and has its other endpoint on the opposite side of that angle.

An angle bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.

In this figure, $\frac{AB}{AC} = \frac{BD}{DC}$

A perpendicular bisector is a segment, ray, or line that intersects a given segment at a 90$^o$, and passes through its mid point.

To construct the perpendicular bisector of a segment follow the following steps:

Step 3

Step 4:

A point lies on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.

In this figure: AH = BH, AG = BG, AF = BF and AE = BE.

The equation of the perpendicular bisector of the segment joining the points ($x_1, y_1$) and ($x_2, y_2$).

Equation of $\bar{RM}$ is:

y - $\frac{y_1 + y _2}{2}$ = - $\frac{x_2 - x_1}{y_2 - y _1}$ (x - $\frac{x_1 + x _2}{2}$)

The slope of perpendicular bisector = Negative reciprocal of line segment's slope.

The perpendicular bisectors of the sides of a triangle intersect at a point. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.

If a point is on the bisector of an line, then it is equidistant from the endpoints of the segment.

Find the equation of the perpendicular bisector of the line segment joining the points (1, 2) and (3, 6).

Step 1:

Slope of the line segment joining the points (1, 2) and (3, 6)

Slope (m) = $\frac{6-2}{3-1}$ = $\frac{4}{2}$ = 2

Mid point: ($\frac{1+3}{2}$, $\frac{2+6}{2}$) = (2, 4)

Step 2:

Step 3:

y - 4 = $\frac{-1}{2}$(x - 2)

2y - 8 = -x + 2

2y = -x + 10

or y = -0.5x + 5, which is required equation.

Given below are some of the practice problems on perpendicular bisector.

$\bar{CA} = 46, \bar{CD} = 49, \bar{BA} = 20, \bar{DB} = ?$

The given sides are $\bar{CA} = 46, \bar{CD} = 49$ and $\bar{BA} = 20$

So, we have to find out $\bar{DB}$

Using the angle bisector theorem, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

$\frac{46}{49}$ = $\frac{20}{\bar{DB}}$

$46 \bar{DB} = 980$

$\bar{DB} = 21.30$

So, we have to find out $\bar{DB}$

Using the angle bisector theorem, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

$\frac{46}{49}$ = $\frac{20}{\bar{DB}}$

$46 \bar{DB} = 980$

$\bar{DB} = 21.30$

Find the missing side length

$\bar{CA} = 32, \bar{CD} = 57, \bar{BA} = 30$ and $\bar{DB} = ?$

The given sides are $\bar{CA} = 32, \bar{CD} = 57$ and $\bar{BA} = 30$

So, we have to find out $\bar{DB}$

Using the angle bisector theorem, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

$\frac{32}{57}$ = $\frac{30}{\bar{DB}}$

$32 \bar{DB} = 1710$

$\bar{DB} = 53.43$

So, we have to find out $\bar{DB}$

Using the angle bisector theorem, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

$\frac{32}{57}$ = $\frac{30}{\bar{DB}}$

$32 \bar{DB} = 1710$

$\bar{DB} = 53.43$

More topics in Bisector | |

Angle Bisector | Bisector of a Line |

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