Topics
Similarity
- Properties of Ratios of Areas of Two Triangles
- Basic Proportionality Theorem
- Property of an Angle Bisector of a Triangle
- Property of Three Parallel Lines and Their Transversals
- Similarity of Triangles (Corresponding Sides & Angles)
- Relation Between the Areas of Two Triangles
- Criteria for Similarity of Triangles
- Overview of Similarity
Pythagoras Theorem
- Pythagoras Theorem
- Pythagorean Triplet
- Property of 30°- 60°- 90° Triangle Theorem
- Property of 45°- 45°- 90° Triangle Theorem
- Similarity in Right Angled Triangles
- Theorem of Geometric Mean
- Right-angled Triangles and Pythagoras Property
- Converse of Pythagoras Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Apollonius Theorem
- Overview of Pythagoras Theorem
Circle
- Circles Passing Through One, Two, Three Points
- Tangent and Secant Properties
- Secant and Tangent
- Inscribed Angle Theorem
- Intersecting Chords and Tangents
- Corollaries of Inscribed Angle Theorem
- Angle Subtended by the Arc to the Point on the Circle
- Angle Subtended by the Arc to the Centre
- Overview of Circle
Geometric Constructions
Co-ordinate Geometry
Trigonometry
- Trigonometric Ratios in Terms of Coordinates of Point
- Angles in Standard Position
- Trigonometric Ratios
- Trigonometry Ratio of Zero Degree and Negative Angles
- Trigonometric Table
- Trigonometric Identities (Square Relations)
- Angles of Elevation and Depression
- Relation Among Trigonometric Ratios
- Trigonometric Ratios of Specific Angles
Mensuration
Definition: Pythagorean Triplet
Three natural numbers (a,b,c) form a Pythagorean triplet if:
c2 = a2 + b2
(where c is the largest number)
Examples:
(3,4,5), (5,12,13), (8,15,17)
Formula: Pythagorean Triplets
If a > b, then:
(a2 + b2, a2 − b2, 2ab) is a Pythagorean triplet.
Key Points: Properties of Right Angled Triangle with the Angles
(I)Property of 30°-60°-90° triangle
- Side opposite 30° = \[\frac{1}{2}\] × hypotenuse
- Side opposite 60° = \[\frac{\sqrt{3}}{2}\] × hypotenuse.
(II) Property of 45°-45°-90°
- Each perpendicular side = \[\frac{1}{\sqrt{2}}\] × hypotenuse
Theorem: Similarity and Right Angled Triangle
Statement:
In a right-angled triangle, if the altitude is drawn to the hypotenuse, then the two triangles formed are similar to the original triangle and to each other.
△ADB ~ △ АBС
△ BDC ~ △ ABC
△ ADB ~ △ BDC
Theorem: Theorem of Geometric Mean
Statement:
In a right-angled triangle, the altitude drawn to the hypotenuse is the geometric mean of the two segments of the hypotenuse.
(Altitude)2 = (segment1) × (segment2)
Theorem: Pythagoras Theorem
Statement:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Hypotenuse2 = (base)2 + (perpendicular)2
Converse of Pythagoras Theorem:
In a triangle, if the square of one side is equal to the sum of the squares of the remaining two sides, then the triangle is a right-angled triangle
Theorem: Apollonius Theorem
Statement:
In a triangle, the sum of the squares of any two sides is equal to twice the square of the median to the third side plus twice the square of half the third side.
AB2 + AC2 = 2AM2 + 2BM2
(where M is the midpoint of BC)
