Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry → Constructions ↓


Construction of Angle Bisectors


Constructions in geometry often reflect the beauty and precision of mathematical principles. One of the fundamental problems is to construct an angle bisector. An angle bisector is a line or ray that divides an angle into two equal parts. Let's see step by step how to construct an angle bisector and understand the geometry behind it.

Angle bisectors are fascinating because they maintain the shortest path from a point to a line. Understanding this concept is essential for a variety of applications within geometry, engineering, architecture, and even computer graphics. Let's take a deeper look at the steps and the logic behind each step.

Steps to construct an angle bisector

The following guide shows you how to construct an angle bisector using only a compass and a straight line:

  1. Draw the angle: Start by drawing an arbitrary angle, call it ∠ABC, where point B is the vertex of the angle. Here's how you typically represent it:
         A
     ,
      ,
       B ------------ C
  2. Place the tip of the compass at the vertex: Place the tip of the compass at the vertex B of the angle. Adjust the compass to a convenient width. This width is arbitrary but must remain the same during the next two steps.
  3. Draw an arc on both rays: With the compass set to your chosen width, draw an arc on either side of the angle (AB and BC). Let's say the points where it intersects the rays are D and E. BACDI
  4. Place the compass point at the intersection points: Without changing the compass width, place the compass at point D and draw an arc within the angle. Keep the compass width the same and place the compass point at point E, then draw another arc within the angle. Let the two arcs intersect at F.
  5. Draw the bisector: Use a straightedge to draw a line from the vertex B through the intersection point F of the arcs in the previous step. This line BF is the angle bisector, which means it divides ∠ABC into two equal angles. BACDIF

With these steps, you can divide any angle into two equal parts using only a compass and a straight line. The beauty of this technique lies in its simplicity and accuracy. It is an essential tool for engineers, architects, and mathematicians.

Understanding the geometry

The method of constructing an angle bisector is deeply rooted in the principles of Euclidean geometry. Let's analyze why this method works in different scenarios and is reliable.

When you draw an arc from the vertex that intersects the rays of the angle, you essentially create two congruent line segments (BD and BE in this case). This step lays the groundwork for accurately finding the midpoint of the angle.

The arcs coming from the intersection create a unique point (F) that lies on the angle bisector. This is because the arcs coming from D and E are of the same radius, ensuring that point F is equidistant from both D and E.

Mathematical explanation

From a mathematical point of view, constructing an angle bisector involves several important concepts:

  • Similar line segments: Two arcs form segments BD and BE, which are similar by structure.
  • Perpendicular bisector: Point F is equidistant from D and E, which means BF is the locus of points equidistant from D and E.
  • Angle equivalence: By construction, ∠ABF is equilateral to ∠CBF, hence BF is the bisector of ∠ABC.

Applications of angle bisector

The construction of angle bisectors is used in many areas of mathematics and beyond. Here are some scenarios where angle bisectors play an important role:

  • Incentre of a triangle: The angle bisectors of a triangle meet at a point called the incentre, which is the centre of the circle inscribed within the triangle.
  • Minimum path problems: Angle bisectors are used to find the minimum path between a point and lines (or surfaces in extended cases).
  • Design and architecture: Angle bisectors help ensure symmetry in symmetrical designs and in architectural constructions.
  • Physics and optics: They are used in optics, where angle bisectors are important in understanding light paths and refraction.

Examples of bisector construction

Example 1: Straight angles

Consider constructing an angle bisector for a simple angle. Let's say you have an angle of 60 degrees. Using the above steps, you can divide it into two angles of 30 degrees. The construction process will be exactly the same as described earlier, working perfectly regardless of the initial angle.

Example 2: Practical application

Imagine you are in a surveying context where you need to divide the land into two equal parts starting from a point at an angle. By using the angle bisector construction, you can ensure that the equal division automatically respects the angular measure of the starting position.

Conclusion

Constructing angle bisectors is a foundational skill in geometry, essential not only for academic purposes but also for practical applications. These steps, based on the rigorous logic of Euclidean geometry, will allow you to handle angles precisely. Understanding and mastering angle bisector construction not only enhances your geometric skills but enriches your appreciation for the underlying harmony and symmetry found in mathematics.


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