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# Bisector

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.

## Bisector Definition

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.

## Segment Bisector

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

### Line Segment Bisector

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.

## Angle 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$^o$.

## Triangle Bisector

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, $\frac{AB}{AC} = \frac{BD}{DC}$

## Perpendicular Bisector

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).

## Bisector Examples

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:
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, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

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

$46 \bar{DB} = 980$

$\bar{DB} = 21.30$

Question 2:

Find the missing side length

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

Solution:
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, $\frac{\bar{CA}}{\bar{CD}}$ = $\frac{\bar{BA}}{\bar{DB}}$

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

$32 \bar{DB} = 1710$

$\bar{DB} = 53.43$

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