9th Class Math Chapter 12 Solution | Get Now

9th Class Math Chapter 12 Solution covers Line Bisectors and Angle Bisectors, an important geometry chapter in the Punjab Board syllabus. This chapter explains the properties of right bisectors and angle bisectors, along with their practical applications in real-life problems like locating equidistant points.

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Students preparing for exams need a clear 9th Class Math Chapter 12 Solution to understand how these theorems are proved and applied in different exercise questions.

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What is a Right Bisector of a Line Segment?

A line is called a right bisector of a line segment if it is perpendicular to the segment and passes through its mid-point. This definition forms the base of the first theorem explained in the 9th Class Math Chapter 12 Solution.

What is the Bisector of an Angle?

A ray BP is called the bisector of angle ABC if P is a point in the interior of the angle and angle ABP equals angle PBC. This concept is used throughout the chapter to prove theorems related to angle bisectors.

Key Theorems in 9th Class Math Chapter 12 Solution

This chapter includes four major theorems that connect right bisectors and angle bisectors with triangle and circle properties.

Theorem 1: Right Bisector Equidistance

Any point on the right bisector of a line segment is equidistant from its endpoints. This is proved using the SAS postulate by joining a point P on the bisector to both endpoints of the segment.

Theorem 2: Converse of Right Bisector

Any point equidistant from the endpoints of a line segment lies on the right bisector of that segment. The 9th Class Math Chapter 12 Solution proves this using the SSS postulate and supplementary angles.

Theorem 3: Concurrency of Right Bisectors

The right bisectors of the sides of a triangle are concurrent, meaning they meet at a single point. This point plays an important role in circle geometry, especially in finding the center of a circumscribed circle.

Theorem 4: Angle Bisector Equidistance

Any point on the bisector of an angle is equidistant from its arms. This is proved using the SAA postulate by drawing perpendiculars from the point to both arms of the angle.

Concurrency Properties Explained

The 9th Class Math Chapter 12 Solution also explains an important theorem stating that the bisectors of the angles of a triangle are concurrent. This point of concurrency is equidistant from all three sides of the triangle.

Types of Triangle Intersections

Understanding where right bisectors meet depends on the triangle type:

• In an acute triangle, the right bisectors intersect inside the triangle
• In a right triangle, the right bisectors intersect on the hypotenuse
• In an obtuse triangle, the right bisectors intersect outside the triangle

This concept is frequently tested in objective-type questions across board exams.

Real-Life Application of Bisectors

One of the most practical examples in the 9th Class Math Chapter 12 Solution involves finding a location equidistant from three villages. This type of question demonstrates how right bisectors are used to solve real-world placement problems.

Example Problem

If three villages P, Q, and R are not on the same line, and the people want to build a park equidistant from all three villages, the solution involves:

1. Completing triangle PQR by joining the villages
2. Drawing right bisectors of two sides
3. Finding their point of intersection
4. Proving this point is equidistant from all three villages using SAS postulate

This same principle applies to finding the center of a circle passing through three non-collinear points.

Exercise-Wise Breakdown

The 9th Class Math Chapter 12 Solution is divided into two main exercises along with an objective section:

• Exercise 12.1 – Covers right bisector theorems, circle center problems, and the equidistant park example
• Exercise 12.2 – Focuses on angle bisector proofs, quadrilateral bisector problems, and triangle bisector concurrency
• Objective Section – Includes true/false questions and numerical problems on bisectors

Sample Problem from Exercise 12.1

A common question asks students to prove that the center of a circle lies on the right bisectors of each of its chords. This is solved using the HS postulate by drawing a perpendicular from the center to the chord.

Sample Problem from Exercise 12.2

Another frequent question involves proving that if the right bisectors of two sides of a quadrilateral meet at a point, that point lies on the bisector of a specific angle. The 9th Class Math Chapter 12 Solution solves this using the SSS postulate.

Numerical Problems on Bisectors

This chapter also includes numerical questions where students calculate unknown angles and lengths using bisector properties. For example, in an equilateral triangle where AD bisects angle A:

• Each angle of the triangle equals 60°
• The bisector divides the angle into two equal parts of 30° each
• The base angle formed at point D equals 90°, since AD is also the right bisector of the base

These types of problems help students understand how bisector theorems apply to specific triangle types.

Tips for Solving Bisector Proofs

Students working through the 9th Class Math Chapter 12 Solution should follow this approach for accuracy:

1. Identify whether the question involves a line bisector or an angle bisector
2. Write the “Given” and “To Prove” statements clearly
3. Add necessary construction, such as joining points or drawing perpendiculars
4. Choose the correct postulate (SAS, SSS, or SAA) based on available information
5. Conclude with the required equidistance or bisector proof

Following this method ensures students present proofs in the format expected by board examiners.

Suggested Image Alt Text: 9th Class Math Chapter 12 Solution – Line Bisectors and Angle Bisectors Proof Diagram

FAQs

Q1. What does 9th Class Math Chapter 12 Solution cover?
It covers Line Bisectors and Angle Bisectors, including theorems on right bisector equidistance, angle bisector properties, and concurrency of bisectors in triangles with solved exercises.

Q2. What is a right bisector of a line segment?
A right bisector is a line that is perpendicular to a line segment and passes through its mid-point. Any point on this bisector is equidistant from both endpoints of the segment.

Q3. Where do the right bisectors of a triangle intersect?
In an acute triangle, they intersect inside; in a right triangle, they intersect on the hypotenuse; and in an obtuse triangle, they intersect outside the triangle.

Q4. What is the bisector of an angle used for?
An angle bisector divides an angle into two equal parts. Any point on the bisector is equidistant from both arms of the angle, which is proved in the 9th Class Math Chapter 12 Solution.

Q5. How many exercises are in Chapter 12 of 9th class math?
Chapter 12 contains two exercises (12.1 and 12.2), covering right bisector theorems and angle bisector proofs, along with an objective section of true/false and MCQ questions.

Q6. Where can I find 9th Class Math Chapter 12 Solution for free?
A complete 9th Class Math Chapter 12 Solution with solved theorems, exercises, and objective questions is available for free download on TaleemWorld.com for exam preparation.