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
Theorem: Opposite angles of a cyclic quadrilateral are supplementary
Statement:
The sum of the opposite angles of a cyclic quadrilateral is 180°.
Short Proof (Idea):
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Let ABCD be a cyclic quadrilateral.
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Arc ABC subtends an angle ∠ADC at the circle and ∠AOC at the centre.
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The angle at the centre is double the angle at the circle.
∠ADC=`1/2`∠AOC
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Similarly, the other arc subtends:
∠ABC = `1/2`(reflex ∠AOC)
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The sum of angles around the centre is 360°.
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Therefore,
∠ADC + ∠ABC = `1/2`(360°) = 180∘
Conclusion:
Hence, the opposite angles of a cyclic quadrilateral are supplementary.
Theorem: Converse of Cyclic Quadrilateral
Statement:
If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.
Short Proof (Idea):
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Given, ∠B + ∠D = 180°.
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Draw a circle through three vertices of the quadrilateral.
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If the fourth vertex does not lie on the circle, an exterior angle becomes equal to its interior opposite angle, which is not possible.
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Hence, the fourth vertex must lie on the same circle.
Conclusion:
Therefore, ABCD is a cyclic quadrilateral.
Theorem: Exterior Angle of a Cyclic Quadrilateral
Statement:
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
Short Proof (Idea):
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In a cyclic quadrilateral, the sum of opposite angles is 180°.
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The exterior angle and the adjacent interior angle form a straight line, so their sum is 180°.
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Since both are supplementary to the same angle, they are equal.
Conclusion:
Therefore, the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
