Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Geometry → Constructions ↓


Constructing Bisectors


In geometry, the concept of a bisector is a cornerstone, especially when you're dealing with angles and line segments. A bisector cuts something into two equal parts. The two main types of bisectors you'll encounter are angle bisectors and perpendicular bisectors of line segments. Understanding how these bisectors are constructed is essential for problem-solving and proofs in geometry. This article dives into the intricate details of constructing bisectors, including step-by-step guides and visual representations.

What is a bisector?

A bisector is a line, segment, or ray that divides another geometric figure into two equal parts. There are two main types of bisectors in geometry:

  • Angle Bisector: This bisector will divide an angle into two equal angles.
  • Perpendicular Bisector: This bisector will cut a line segment into two equal lengths and will be perpendicular to the original line segment.

Construction of angle bisectors

Angle bisectors are important because they ensure that the two newly formed angles are congruent. Here's how you can construct an angle bisector using a compass and straightedge. Follow these steps to construct the angle bisector of ∠ABC:

1. Place the compass at the vertex point B of the angle.
2. Draw an arc on either side of the angle (these are rays BA and BC). Label the crossing points as D and E.
3. Without changing the compass width, place the compass at point D and draw an arc within the angle.
4. Without changing the compass width, place the compass at point E and draw another arc that intersects the first arc. Label the intersection point F.
5. Draw ray BF. This is the angle bisector. It divides ∠ABC into two equal angles: ∠ABF and ∠CBF.
C A B F

The advantage of constructing an angle bisector using the given method is that it does not require knowledge of the actual angle measure involved. The compass and straightedge method ensures that the angle is accurately divided into two parts.

Constructing a perpendicular bisector

The perpendicular bisector is a line that divides another line into two equal segments and forms a right angle (or 90 degrees) with it. Here is a step-by-step guide for constructing the perpendicular bisector of segment AB:

1. Place the compass at point A and adjust its width to more than half the length of AB. Draw an arc above and below the line.
2. Without adjusting the compass width, place the compass at point B and draw a second arc that intersects the first arc at two points. Label these points as P and Q.
3. Draw the line PQ. PQ is the perpendicular bisector of AB as it cuts AB at its midpoint and makes an angle of 90 degrees.
A B

The perpendicular bisector ensures that two equal segments are created without needing to know the initial length of AB. The use of a compass and straightedge ensures high accuracy.

Practical uses of the bisector

In geometry, the construction of bisector has various practical uses, such as

  • Bisecting a line segment: Although it is impossible to construct an exact bisector using only a compass and straight line, angle bisectors and perpendicular bisectors provide important steps in arriving at a bisector.
  • Constructing medians and altitudes: Perpendicular bisectors help in finding medians and altitudes in triangles, which are needed in determining the center of gravity and perpendicular centers respectively.

More examples of construction

Let us explore more cases of angle bisector and perpendicular bisector construction through examples:

Example 1: Dividing an angle in different ratios

Sometimes, angles must be divided into non-equal parts. For example, a 3:2 ratio can be obtained with iterative construction:

1. Start with the steps to construct angle bisector BD of ∠ABC.
2. Draw the perpendicular bisector of BD which meets it at G.
3. Draw GH parallel to BC. Produce the other side F of the angle forming ∠BGD in the desired ratio of 3:2 with practical measurements and construction.

This repeated use of bisectors and practical adjustments can solve various complex geometrical problems involving ratios and angle division. Measurements through practical adjustments and constructions, visualization through figures further validates the accuracy.

Example 2: Constructing an isosceles triangle with a given base and height

Angles and perpendicular bisectors come in handy here:

1. Draw the base AB of the desired length.
2. Construct a perpendicular bisector line CD of AB, with C at the desired height on the bisector.
3. Connect points A and B with C.

Conclusion

Learning how to construct bisectors is essential to mastering geometry. These fundamental skills lay the groundwork for more complex geometric problem-solving. Learning to properly use a compass and straightedge strengthens accuracy and precision in constructions, which is beneficial in both academic and practical fields. Whether you're solving problems or proving theorems, mastering bisectors is invaluable.


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Constructing Bisectors

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