To define, bisection is dividing a shape into two congruent parts, and the line segment that helps to divide the shape into congruent parts is known as the bisector. In general, a bisector is a line, and the common types of bisectors are perpendicular bisector of a line segment and angle bisector.
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From the above picture, the following conclusions can be made:
Line AB is the perpendicular bisector of line segment ED
Line BD is the bisector of line EF at point D
Line AB is the interior bisector of angle EBD
Did You Know?
Bisection, in 3D space, is done using a plane which is also denoted as a bisecting plane.
Next, let’s proceed with the types of the bisector.
Angle Bisector
Theorem:
The Angle Bisector theorem explains that an angle bisector of a triangle will divide the opposite side into two parts which are in proportion with the other two side arms of the triangle. Let’s see an example.
Draw a triangle ABC and an angle bisector from A that cuts the side BC at point D. Considering the theorem, the following equation can be achieved:
|BD| / |DC| = |AB| / |AC|
Furthermore, a generalized version of the angle bisector theorem explains that if point D is on the line segment BC, we can express it as:
|BD| / |DC| = |AB| sin DAB / |AC| sin DAC
This form can be reduced to the previous equation when AD becomes bisector of angle BAC. In general, this angle bisector hypothesis has extensive usage when values of side lengths and angle bisector are provided.
In the following section, we have provided some proofs regarding the angle bisector theorem.
Considering the previous diagram, the law of sines is used on the triangles ACD and ABD:
|AB| / |BD| = sin BDA / sin BAD ..........(i)
|AD| / |DC| = sin ADC / sinDAC ..........(ii)
The two angles ADC and BDA constitute to be a linear pair which means they are side by side supplementary angles. Also, all supplementary angles have similar sines, hence:
sin BDA = sin ADC
As these two angles are the same, the right-hand portion of equations (i) and (ii) are also the same. So, the left portions are also similar.
|BD| / |DC| = |AB| / |AC|
Solved Numerical
Question: Consider a triangle ABC having an angle bisector AD on the portion BC. If the length of BD, CD, and AB + AC is 2, 5, and 10, respectively, determine AB and AC.
Solution: Following the angle bisector theorem, it can be written as AB/2, AC/5, or AB = 2/5AC. Putting the values in AB + AC = 10, we get AC = 50/7. Similarly, you can substitute values to find AB as well, which are found out to be 20/7.
Line Segment Bisector
With regards to a bisector of a segment, it is nothing but a point that cuts a line into two equal parts. As shown in the below figure, the line AB is divided into parts having the same lengths.
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Moreover, if a line bisector makes an angle of 90 degrees while intersecting line AB, it is considered a perpendicular bisector. On the other hand, if a line bisector intersects at a different angle, it is denoted only as a bisector.
How to Construct a Perpendicular Bisector?
Here, we have provided a step by step procedure of how to draw and find perpendicular bisector.
First, draw a line and name the two endpoints P and Q.
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Next, put your compass on point P and set it at a distance of more than two-thirds portion of the line segment PQ. Note that precise width is not required.
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After that draw two arcs above and below the line PQ.
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Again from point Q, repeat the same thing.
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Now, at the intersection points of the two arcs, mark two points and draw a line joining them.
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Do It Yourself
1. When you construct the angle bisectors of a triangle, a concurrency point is created. What is it called?
(a) Incenter (b) Centroid (c) Orthocenter (d) Circumcenter
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FAQs on Bisector
1. What are the Properties of a Perpendicular Bisector?
Ans. Firstly, a perpendicular bisector cuts a line segment into two equal lengths. Secondly, the bisector forms right angles with the line. Thirdly, each and every point on the bisector is equidistant from the two ends of the line segment.
2. What is the Dissimilarity Between the Perpendicular Bisector Theorem and the Angle Bisector Theorem?
Ans. Many a time, students get confused about the two theorems– perpendicular and angle bisector, as they look almost similar. However, conclusions drawn from the two theorems are entirely different. The angle bisector theorem talks about angles of congruency, which creates equal distances from the center to the side of a triangle. On the other hand, the perpendicular bisector theorem of a triangle is concerned with a triangle’s congruent portion. This allows congruency between the diagonals from the triangle arms to the circumcenter.
3. What is the Expression for a Perpendicular Bisector of a Line Joining Points (3,5) and (7,1)?
Ans. A perpendicular bisector, M can be considered as the midpoint on the line suppose PQ. So, it can be written as midpoint (PQ) = [ 7+3/2 , 5+1/2] = [10/2 , 6/2] = M [5, 3].
Further, slope (PQ) m1 = 3 – 7/ 5 – 1 = -4/1 = -1; m1 x m2 = -1; -1 x m2 = -1, therefore m2 = 1.
Finally, expression for this perpendicular bisector can be written as (y – 3) = 1(x – 5), which is y - 3 = x - 5.