Types of Bisectors in Geometry with Formulas and Applications
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 Maths: Meaning, Formula, Types & Solved Examples
1. What is binary to decimal conversion in maths?
In mathematics, binary to decimal conversion involves changing a binary number (base 2, using only 0 and 1) into its equivalent decimal number (base 10, using digits 0-9). This is crucial for understanding how computers represent and process information.
2. What is the formula for converting binary to decimal?
The formula for binary to decimal conversion is: Decimal = (dn × 2n) + (dn-1 × 2n-1) + ... + (d1 × 21) + (d0 × 20), where di represents each digit in the binary number and n is the position of the digit (starting from the rightmost digit, with position 0).
3. How do you convert 10101 from binary to decimal?
To convert 10101 (binary) to decimal:
• 1 × 24 = 16
• 0 × 23 = 0
• 1 × 22 = 4
• 0 × 21 = 0
• 1 × 20 = 1
Adding these results: 16 + 0 + 4 + 0 + 1 = 21 (decimal).
4. Can I convert binary to decimal without a calculator?
Yes, you can! Use the formula and method shown above. For smaller binary numbers, you can perform the calculations mentally. For larger numbers, using pen and paper is recommended for accuracy.
5. Why is binary to decimal conversion important in computer science?
Binary to decimal conversion is fundamental in computer science because computers operate using binary code (0s and 1s). Understanding this conversion is key to interpreting data, comprehending how information is stored and processed, and working with digital circuits.
6. What are common mistakes students make during binary to decimal conversion?
Common errors include:
• Incorrectly identifying place values (powers of 2)
• Miscalculating the powers of 2
• Making addition errors
• Forgetting to include the rightmost digit (20)
7. How does binary to decimal conversion differ for fractional binary numbers?
For fractional binary numbers (those with a decimal point), the place values to the right of the point are negative powers of 2 (2-1, 2-2, etc.). You follow the same multiplication and summation process, but with negative exponents.
8. Are there mental math tricks for faster binary to decimal conversion?
Yes, practice recognizing common binary patterns. For instance, 1000 is 8, 100 is 4, 10 is 2, and 1 is 1. You can quickly sum these values for faster mental calculations.
9. Can binary to decimal conversion be applied to hexadecimal numbers?
While not a direct conversion, you can convert hexadecimal (base 16) to binary first, and then convert the binary representation to decimal. Hexadecimal is often used as a shorthand for binary because each hexadecimal digit represents four binary digits.
10. How is binary to decimal conversion tested in school board exams or Olympiads?
Expect questions ranging from direct conversions of given binary numbers to word problems involving binary representations and their decimal equivalents. Sometimes, you might need to apply the concept in problem-solving related to computer science or digital electronics.
11. What is the decimal equivalent of the binary number 110110?
To convert 110110 (binary) to decimal:
• 1 × 25 = 32
• 1 × 24 = 16
• 0 × 23 = 0
• 1 × 22 = 4
• 1 × 21 = 2
• 0 × 20 = 0
Adding these results: 32 + 16 + 0 + 4 + 2 + 0 = 54 (decimal).
12. Explain the significance of Most Significant Bit (MSB) and Least Significant Bit (LSB) in binary to decimal conversion.
The MSB is the leftmost digit in a binary number, holding the highest positional value (the largest power of 2). The LSB is the rightmost digit, representing the lowest positional value (20). Understanding MSB and LSB is crucial for correctly applying the conversion formula and avoiding errors.
