Topics
Number Systems
Number Systems
Algebra
Introduction to Polynomials
Algebraic Expressions
Algebraic Identities
Sequences and Progressions
Coordinate Geometry
Geometry
Exploring Algebraic Identities
Area
Constructions
- Introduction of Constructions
- Geometric Constructions
- Some Constructions of Triangles
Mensuration
Linear Equations in Two Variables
Statistics and Probability
Coordinate Geometry
Probability
Introduction to Euclid’s Geometry: Axioms and Postulates
Lines and Angles
- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Concept of Pairs of Angles
- Concept of Transversal Lines
- Basic Properties of a Triangle
Triangles: Congruence Theorems
4-gons (Quadrilaterals)
- Properties of Quadrilateral
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
Circles
Area and Perimeter
- Area of a Triangle by Heron's Formula
- Application of Heron’s Formula in Finding Areas of Quadrilaterals
- Geometric Interpretation of the Area of a Triangle
Surface Area and Volume
Statistics
Introduction to Probability
- Collinearity of three points
Notes
Heron was born in about 10AD possibly in Alexandria in Egypt. He worked in applied mathematics. His geometrical works deal largely with problems on mensuration written in three books. In this book, Heron has derived the famous formula for the area of a triangle in terms of its three sides.
The formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as:
| Area of a triangle = `sqrt (s(s-a) (s - b)(s-c))` |
where a, b and c are the sides of the triangle, and s = semi-perimeter, i.e., half the perimeter of the triangle = `(a + b + c) /2`,
For instances, Let us take a = 40 m, b = 24 m, c = 32 m,
so that we have s = `(40 + 24 + 32 )/ 2` m
= 48 m
s - a = (48 - 40) m = 8m
s - b = (48 - 24)m = 24 m
s - c = (48 - 32 ) m = 16 m
Therefore, area of the park ABC
`= sqrt (s(s - a) (s - b) (s - c))`
`= sqrt (48 * 8 * 24 * 16) m^2 = 384 m^2`
