To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)
Top

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

Back to Top
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

Back to Top

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.

Line 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

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.
Angle Bisector Problem

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

Triangle Bisector

Back to Top
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.

Bisector theorem

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

Perpendicular Bisector

Back to Top
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.

How to Construct Perpendicular Bisector

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.

Perpendicular Bisector Theorem Proof

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 Perpendicular Bisector

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.
Bisectors of 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

Back to Top
Given below are the examples on bisector.

Solved Examples

Question 1: Find the missing side length

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

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

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

$32 \bar{DB} = 1710$

$\bar{DB} = 53.43$

More topics in Bisector
Angle Bisector Bisector of a Line
NCERT Solutions
NCERT Solutions NCERT Solutions CLASS 6 NCERT Solutions CLASS 7 NCERT Solutions CLASS 8 NCERT Solutions CLASS 9 NCERT Solutions CLASS 10 NCERT Solutions CLASS 11 NCERT Solutions CLASS 12
Related Topics
Math Help Online Online Math Tutor
*AP and SAT are registered trademarks of the College Board.