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
CISCE: Class 10
Points Equidistant from Two Intersecting Lines
Statement:
The locus of a point equidistant from two intersecting lines is the pair of angle bisectors of the angles formed by the given lines.
Given:
- Lines AB and CD intersect at O
- P is a point inside ∠BOD
- PM ⟂ OB, PN ⟂ OD
- PM = PN
To Prove:
-
P lies on the bisector of ∠BOD
Construction:
-
Join OP
Proof:
- In ΔOMP and ΔONP, we have
(i) PM = PN (given)
(ii) ∠OMP = ∠ONP = 90° (PM ⟂ OB, PN ⟂ OD)
(iii) OP = OP (common) -
∴ ΔOMP ≅ ΔONP (RHS congruence)
-
∴ ∠MOP = ∠PON (c.p.c.t.)
- Hence, OP bisects ∠BOD.
Conclusion:
Therefore, P lies on the bisector of ∠BOD.
CISCE: Class 10
Converse: Points Equidistant from Two Intersecting Lines
Statement:
Every point on the angle bisector of two intersecting lines is equidistant from the lines.
Given:
- Lines AB and CD intersect at O
- OE is the bisector of ∠BOD
- P is a point on OE
- PM ⟂ OB and PN ⟂ OD
To Prove:
PM = PN
Proof:
-
In ΔOMP and ΔONP, we have
(i) ∠MOP = ∠PON (OE bisects ∠BOD)
(ii) ∠OMP = ∠ONP = 90° (PM ⟂ OB, PN ⟂ OD)
(iii) OP = OP (common) -
∴ ΔOMP ≅ ΔONP (ASA congruence)
-
∴ PM = PN (c.p.c.t.)
Conclusion:
Hence, P is equidistant from OB and OD, and therefore from the intersecting lines AB and CD.
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