Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Constructions ↓


Constructing Bisectors


Welcome to the fascinating world of geometry! In this article, we will explore the concept of constructing bisectors. Bisectors are fundamental in geometry, and understanding how to construct them helps us appreciate the beauty of mathematical precision. Let's dive in!

Understanding the bisector

In geometry, a bisector refers to a line, ray, or segment that divides another figure into two equal parts. The most common types of bisectors include line segment bisectors and angle bisectors. Our main focus will be to learn how to accurately construct these bisectors using basic tools like a ruler and compass.

Tools for construction

To successfully build a bisector, you'll need a few essential tools:

  • A compass, which helps draw arcs and circles.
  • A ruler or straight line, used for drawing straight lines.
  • A pencil, for marking points and drawing lines.

Constructing a line segment bisector

Let us first learn how to construct a line segment bisector. A line segment bisector is a line or segment that cuts another line segment into two equal parts and forms a right angle (90 degrees) with it.

Example: Bisecting a line segment

Suppose you have a line segment AB. Here is how to bisect it:

  1. Place the compass point at one end of line segment A, and draw an arc on either side of the line segment.
  2. Without changing the width of the compass, repeat the same steps from the other endpoint B. The arcs will intersect at the top and bottom of the line segment.
  3. Mark the points of intersection of the arcs with C and D.
  4. Use a ruler to draw a straight line through these points C and D.
  5. This line segment CD is the perpendicular bisector of AB.
    SVG representation of line AB and its bisecting line CD:
    
    A
    B
    
    
    
    C
    D

Constructing an angle bisector

Next, we will learn how to construct an angle bisector. An angle bisector is a line that divides an angle into two equal angles. This construction is useful for solving various problems in geometry where the angles must be equal.

Example: Bisecting an angle

Suppose you have an angle ∠ABC. Follow these steps to bisect it:

  1. Place the compass point at the vertex B of the angle.
  2. Draw an arc intersecting both sides of the angle AB and BC. Mark the intersection points with X and Y.
  3. Keeping the compass on X, draw an arc inside the angle.
  4. Repeat the same arc from point Y, making sure that both arcs intersect each other. Mark the intersection point with Z.
  5. Draw a straight line from vertex B to point Z.
  6. This line BZ is the bisector of angle ∠ABC.
    SVG representation of angle ∠ABC and its bisecting line BZ:
    
    
    A
    C
    B
    
    
    
    Jade

Why is it important to construct a bisector

Constructing a bisector is important in many areas of geometry. This is because:

  • Symmetry and Balance: Bisectors help establish symmetry by ensuring that shapes and angles are divided evenly, which helps to understand balance in design.
  • Medians in Triangles: The concept of angle bisectors is important when working with medians or finding the point where they intersect, which is the centroid of the triangle.
  • Measurement and Properties: With the help of bisectors we can get accurate measurements which are otherwise complicated, and learn more about the properties of various geometric shapes.
  • Problem Solving: Bisectors are often used in mixed geometric problems where the lengths of segments or angle values need to be equal.

Practice makes perfect

Practice is essential to master bisector construction. Here are some exercises you can try yourself:

Exercise 1: Bisecting the angles of a triangle

Draw a triangle and bisect each of its interior angles. Confirm whether the bisectors meet at a point, known as the incenter. This point is the center of the inscribed circle of the triangle.

Exercise 2: Perpendicular bisector in a quadrilateral

Draw any quadrilateral and choose a pair of opposite sides. Draw perpendicular bisectors for these pairs and see if they intersect each other.

Final thoughts

Understanding and constructing bisectors is a fundamental skill in geometry that enhances spatial reasoning and problem-solving abilities. Whether you are bisecting lines or angles, the techniques remain largely the same and provide a foundation for exploring more complex geometric concepts. Enjoy building!


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Constructing Bisectors

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