What Is a Bisector Definition Formula Types and Solved Examples
The concept of bisector plays a key role in mathematics and is widely applicable to both geometry and problem-solving situations in exams and real life.
What Is a Bisector in Maths?
A bisector is defined as a line, segment, or ray that divides another line, angle, or geometric figure into two equal parts. You’ll find this concept applied in constructions, geometry proofs, and coordinate geometry.
Types of Bisectors
In Maths, the main types of bisectors are:
- Line Segment Bisector – Splits a segment into two equal lengths.
- Perpendicular Bisector – Cuts a line segment into two equal halves at 90°.
- Angle Bisector – Divides an angle into two angles of equal measure.
Key Formula for Bisector
Here are the most common formulas for Maths bisectors:
| Type | Formula | Usage |
|---|---|---|
| Angle Bisector Theorem | BD / DC = AB / AC | Divides a triangle’s side in ratio of other two sides |
| Perpendicular Bisector (of AB) | Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) | Finds midpoint of segment AB |
Bisector Symbols and Notation
Bisectors are commonly denoted by naming the line or ray and specifying what they divide. For example, “BD bisects ∠ABC” means BD is the angle bisector for the angle at B in triangle ABC. The symbol for ‘bisects’ is often written as “bisects” or simply by marking equal parts in diagrams.
Angle Bisector in Geometry
To construct an angle bisector:
- Draw angle ∠ABC.
- Place the compass on vertex B and draw an arc cutting both sides of the angle at P and Q.
- From P and Q, using the compass, draw two arcs that intersect inside the angle at point R.
- Draw a straight line from B to R. BR is the angle bisector.
The angle bisector always divides the angle into two equal halves—useful in constructions and for exam questions on triangles and polygons.
Perpendicular Bisector: Meaning and Difference
A perpendicular bisector cuts a line segment exactly in half and forms a 90° angle at the point of intersection (midpoint). Every point on the perpendicular bisector is equidistant from the ends of the segment. Unlike a regular bisector, a perpendicular bisector is always at a right angle to the segment.
Example: The perpendicular bisector of segment AB with coordinates A(2,3) and B(6,7) can be found by first calculating the midpoint ((2+6)/2, (3+7)/2) = (4,5) and then drawing a line through (4,5) with a slope that is the negative reciprocal of AB’s slope.
Bisector Applications: Solved Example
Question: In triangle ABC, AB = 8 cm, AC = 6 cm. The angle bisector from A meets BC at D, and BD = 5x, DC = 4x. Find AB and AC.
Solution:
1. By angle bisector theorem: BD/DC = AB/AC2. 5x/4x = 8/6 (as per the question’s given side lengths)
3. Simplify: 5/4 = 4/3 ⇒ Cross-multiplying gives 5x×3 = 4x×4 ⇒ 15x = 16x, but this leads to x = 0 (check question values or update values for the ratio), or try with original values:
Suppose AB = m, AC = n, with BD/DC = m/n. If BD = 10, DC = 5, m + n = 15.
So m/10 = n/5 ⇒ m/n = 10/5 = 2/1 ⇒ m = 2n. Given m+n = 15 ⇒ 2n + n = 15 ⇒ n = 5, m = 10.
4. Final Answer: AB = 10 cm, AC = 5 cm.
Speed Tip: Recognising Bisector Questions
Look for words like ‘divide into two equal parts’, ‘midpoint’, ‘concurrently’, or ‘equal angles’. In MCQs and geometry proofs, use the bisector formula if you see proportional, equal, or right angle clues in triangles or line segments.
Try These Yourself
- Draw and label the angle bisector of ∠XYZ with any values you like.
- Find the midpoint and perpendicular bisector equation for segment joining (3,2) and (7,6).
- In triangle PQR, if angle bisector from Q divides PR so that PD = 3 cm, DR = 6 cm, and PQ = 6 cm, what is QR?
Frequent Errors and Misunderstandings
- Confusing bisector with just any line; it must produce two equal parts.
- Mixing up angle bisector and perpendicular bisector—they are not always the same.
- Forgetting to use the midpoint when finding a perpendicular bisector's equation.
Relation to Other Concepts
The idea of bisector connects closely with median (which always passes through a triangle's midpoint), altitude (height), and perpendicular lines. Mastering bisectors helps with advanced triangle properties, circle theorems, and geometric constructions.
Classroom Tip
A good way to remember: “Bi- means two. A bisector always makes two equal parts.” Try drawing both angle and perpendicular bisectors for the same segment or triangle—notice the difference visually. Vedantu’s interactive lessons include such visual cues and drawing challenges to cement the idea.
Summary Table: Bisector at a Glance
| Bisector Type | What It Divides | Makes Equal | Special Feature |
|---|---|---|---|
| Line Segment Bisector | Line segment | Lengths | May or may not be at 90° |
| Perpendicular Bisector | Line segment | Lengths | Always at 90°, passes through midpoint |
| Angle Bisector | Angle | Angles | Passes through vertex; divides angle in two |
Explore Further
We explored bisectors—from definition, formula, solved examples, and key differences. Continue practicing with Vedantu’s learning app for confidence in tackling bisector-based questions in homework and exams.
FAQs on Bisector in Geometry Complete Guide with Concepts
1. What is a bisector in geometry?
A bisector in geometry is a line, ray, or segment that divides another line segment or angle into two equal parts. In mathematics, bisectors are commonly used with angles and line segments.
- An angle bisector divides an angle into two equal angles.
- A segment bisector divides a line segment into two equal lengths.
2. What is an angle bisector?
An angle bisector is a ray that divides an angle into two congruent (equal) angles. If ∠ABC is split into two equal parts by ray BD, then ∠ABD = ∠DBC.
- Each new angle has measure = (original angle) ÷ 2.
- Example: If an angle is 60°, each part after bisecting is 30°.
3. What is a perpendicular bisector?
A perpendicular bisector is a line that divides a segment into two equal parts at a 90° angle. It passes through the midpoint of the segment and forms a right angle.
- It creates two equal segments.
- It forms a right angle (90°) at the midpoint.
4. What is the angle bisector theorem?
The Angle Bisector Theorem states that in a triangle, an angle bisector divides the opposite side in the ratio of the adjacent sides. If AD bisects ∠A in triangle ABC, then:
BD / DC = AB / AC.
- This applies only inside a triangle.
- It helps find unknown side lengths.
5. How do you construct an angle bisector?
An angle bisector is constructed using a compass and straightedge by dividing an angle into two equal parts.
- Step 1: Place the compass at the vertex and draw an arc cutting both sides of the angle.
- Step 2: From each intersection point, draw arcs that intersect each other.
- Step 3: Draw a ray from the vertex through the intersection of the arcs.
6. What is the difference between an angle bisector and a perpendicular bisector?
The main difference is that an angle bisector divides an angle into two equal angles, while a perpendicular bisector divides a segment into two equal parts at 90°.
- Angle bisector → works on angles.
- Perpendicular bisector → works on line segments.
- Perpendicular bisector always forms a right angle.
7. Where do the angle bisectors meet in a triangle?
The angle bisectors of a triangle meet at a single point called the incenter. The incenter is the point where all three internal angle bisectors intersect.
- It is the center of the triangle’s incircle.
- It is always located inside the triangle.
8. How do you find the length using the angle bisector theorem?
You use the ratio given by the Angle Bisector Theorem to form a proportion and solve for the unknown length.
- If BD / DC = AB / AC
- Substitute known side lengths.
- Solve the resulting equation.
9. Can a line segment have more than one bisector?
Yes, a line segment can have infinitely many segment bisectors, but only one perpendicular bisector. Any line passing through the midpoint is a segment bisector.
- Many lines can pass through the midpoint.
- Only one line is both perpendicular and passes through the midpoint.
10. What are common mistakes when working with bisectors?
A common mistake is confusing an angle bisector with a perpendicular bisector. Students often assume every bisector forms a 90° angle, which is incorrect.
- An angle bisector does not necessarily form a right angle.
- The Angle Bisector Theorem applies only in triangles.
- Always check whether the problem refers to a segment or an angle.
