Topics
Compound Interest
- Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
- Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
- Use of Formula
- Finding CI from the Relation CI = A – P
Commercial Mathematics
Goods and Services Tax (G.S.T.)
Banking
Algebra
Geometry
Shares and Dividends
Symmetry
Mensuration
Linear Inequations
Quadratic Equations
- Quadratic Equations
- Method of Solving a Quadratic Equation
- Factorisation Method
- Quadratic Formula (Shreedharacharya's Rule)
- Nature of Roots of a Quadratic Equation
- Equations Reducible to Quadratic Equations
Trigonometry
Statistics
Problems on Quadratic Equations
- Method for Solving a Quadratic Word Problem
- Problems Based on Numbers
- Problems on Ages
- Problems Based on Time and Work
- Problems Based on Distance, Speed and Time
- Problems Based on Geometrical Figures
- Problems on Mensuration
- Problems on C.P. and S.P.
- Miscellaneous Problems
Ratio and Proportion
Probability
Remainder Theorem and Factor Theorem
- Function and Polynomial
- Division Algorithm for Polynomials
- Remainder Theorem
- Factor Theorem
- Applications of Factor Theorem
Matrices
Arithmetic Progression
Geometric Progression
Reflection
- Co-ordinate Geometry
- Advanced Concept of Reflection in Mathematics
- Invariant Points
- Combination of Reflections
- Using Graph Paper for Reflection
Section and Mid-Point Formulae
Equation of a Line
Similarity
Loci
- Locus
- Points Equidistant from Two Given Points
- Points Equidistant from Two Intersecting Lines
- Summary of Important Results on Locus
- Important Points on Concurrency in a Triangle
Angle and Cyclic Properties of a Circle
Tangent Properties of Circles
Constructions
Volume and Surface Area of Solids (Cylinder, Cone and Sphere)
- Mensuration of Cylinder
- Hollow Cylinder
- Mensuration of Cones
- Mensuration of a Sphere
- Hemisphere
- Conversion of Solids
- Solid Figures
- Problems on Mensuration
Trigonometrical Identities
Heights and Distances
- Angles of Elevation and Depression
- Problems based on Elevation and Depression
Graphical Representation of Statistical Data
Measures of Central Tendency (Mean, Median, Quartiles and Mode)
Probability
- Introduction
- Definition: In-Circle
- Definition: Incenter
- Definition: Inradius
- Step-by-Step Construction
- Key Points Summary
Introduction
Imagine you have a triangular piece of land, and you want to build the largest possible circular fountain exactly at the center without crossing the triangle’s boundaries.
The geometry concept that solves this problem is the In-Circle.
This topic is crucial because it connects fundamental geometric principles—specifically angle bisectors—to practical constructions, showing us that every triangle has a unique central point defined by its angles.
Definition: In-Circle
The In-Circle of a triangle is the largest possible circle that can be drawn inside the triangle such that it just touches (is tangent to) all three sides.
Definition: Incenter
The point where all three angle bisectors of a triangle meet. This point is the center of the In-Circle.
Definition: Inradius
The perpendicular distance from the Incenter (I) to any of the three sides. This distance is the radius of the In-Circle.
Step-by-Step Construction
Step 1:
Draw a triangle ABC of any size.
Step 2:
Draw angle bisectors from all three vertices.
Step 3:
Find where they meet → This is the incenter (I).
Step 4:
Determine the Radius (Inradius) → Draw a perpendicular from I to any side → Measure this, it is called the inradius (r).
Step 5:
Draw the In-Circle: Using point I as the centre and length IP as the radius, draw the circle.
Key Points Summary
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In-circle - Circle inside touching all sides.
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Incenter (I) - Center point of In-circle.
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Inradius (r) - Radius of In-circle.
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Angle bisector - Line cutting angle in half.
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Tangent - Line touching circle at one point.
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Tangent point - Where In-circle touches a side.
