Topics
Similarity
- 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)
- Theorem of Areas of Similar Triangles
- Overview of Similarity
Pythagoras Theorem
- Pythagorean Triplet
- Property of 30°- 60°- 90° Triangle Theorem
- Property of 45°- 45°- 90° Triangle Theorem
- Theorem of Similarity in Right Angled Triangle
- Theorem of Geometric Mean
- Pythagoras Theorem
- Application of Pythagoras Theorem
- Apollonius Theorem
- Overview of Pythagoras Theorem
Circle
- Properties of Chords of a Circle
- Circles Passing Through One, Two, Three Points
- Secant and Tangent
- Tangent Theorem
- Tangent Segment Theorem
- Touching Circles
- Arc of the Circle
- Central Angle and the Measure of an Arc
- Congruence of Arcs
- Property of Sum of Measures of Arcs
- Inscribed Angle & Intercepted Arc
- Cyclic Quadrilateral
- Overview of Circle
- Theorem of Angle between Tangent and Secant
- Theorem of Internal Division of Chords
- Theorem of External Division of Chords
- Tangent Secant Segments Theorem
Geometric Constructions
- Construction of Similar Triangle
- Construction of a Tangent to a Circle at a Point on the Circle
- Construction of a Tangent to a Circle from a Point Outside the Circle
- Overview of Geometric Constructions
Co-ordinate Geometry
Trigonometry
Mensuration
Definition: Touching Circles
Two circles are touching if they intersect at exactly one point.
Types:
-
Externally touching circles
Distance between centres = sum of radii -
Internally touching circles
Distance between centres = difference of radii
Key Points: Measure of Arcs
-
An angle whose vertex is the centre of a circle is called a central angle.
-
Measure of minor arc = measure of its central angle
-
Measure of major arc = 360° − minor arc
- The measure of a semicircle is 180°
-
Full circle = 360°
Key Points: Congruence of Arcs
Two arcs are congruent if:
-
They belong to the same or congruent circles
-
They have equal radii
-
They have equal measures
Congruent arcs ⇔ congruent chords
Definition: Inscribed Angle
An inscribed angle is an angle whose vertex lies on the circle and whose arms intersect the circle at two other distinct points.
Definition: Intercepted Arc
The arc of the circle intercepted by the arms of the angle is called the intercepted arc of the inscribed angle.
Theorem: Inscribed Angle Theorem
Statement:
The measure of an inscribed angle is half of the measure of the arc intercepted by it.
Measure of an inscribed angle = `1/2` × measure of intercepted arc
- Angles inscribed in the same arc are congruent.
- An angle inscribed in a semicircle is a right angle.
Key Points: Cyclic Quadrilateral
A cyclic quadrilateral: All four vertices lie on the same circle
Key properties:
-
Opposite angles are supplementary
-
Exterior angle = interior opposite angle
Key points: Angles Formed by Chords, Secants, Tangents
1) Chords intersect inside the circle
Angle = `1/2` (sum of intercepted arcs)
(2) Secants intersect outside the circle
Angle = `1/2`(difference of intercepted arcs)
(3) Tangent–secant angle
Angle = `1/2` (intercepted arc)
These concepts unify angles + arcs.
Theorem: Theorems on Chord and Secant Segments
Chords Intersecting Inside:
AE × EB = CE × ED
Chords Intersecting Outside:
AE × EB = CE × ED
Tangent–secant segments:
EA × EB = ET2
