Topics
Similarity
- Property of three parallel lines and their transversals
- Property of an Angle Bisector of a Triangle
- Basic Proportionality Theorem Or Thales Theorem
- Converse of Basic Proportionality Theorem
- Appolonius Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Similarity
- Properties of Ratios of Areas of Two Triangles
- Similarity of Triangles
- Similar Triangles
- Similarity Triangle Theorem
- Areas of Two Similar Triangles
- Areas of Similar Triangles
Pythagoras Theorem
Circle
- Theorem of External Division of Chords
- Theorem of Internal Division of Chords
- Converse of Theorem of the Angle Between Tangent and Secant
- Theorem of Angle Between Tangent and Secant
- Converse: If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
- Corollary of Cyclic Quadrilateral Theorem
- Theorem: Opposite angles of a cyclic quadrilateral are supplementary.
- Corollaries of Inscribed Angle Theorem
- Inscribed Angle Theorem
- Intercepted Arc
- Inscribed Angle
- Property of Sum of Measures of Arcs
- Tangent Segment Theorem
- Converse of Tangent Theorem
- Circles Passing Through One, Two, Three Points
- Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers
- Cyclic Properties
- Tangent - Secant Theorem
- Cyclic Quadrilateral
- Angle Subtended by the Arc to the Point on the Circle
- Angle Subtended by the Arc to the Centre
- Introduction to an Arc
- Touching Circles
- Number of Tangents from a Point on a Circle
- Tangent to a Circle
- Tangents and Its Properties
- Theorem - Converse of Tangent at Any Point to the Circle is Perpendicular to the Radius
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
Co-ordinate Geometry
Geometric Constructions
- To Construct Tangents to a Circle from a Point Outside the Circle.
- Construction of Triangle If the Base, Angle Opposite to It and Either Median Altitude is Given
- Construction of Tangent Without Using Centre
- Construction of Tangents to a Circle
- Construction of Tangent to the Circle from the Point on the Circle
- Basic Geometric Constructions
- Division of a Line Segment
Trigonometry
Mensuration
notes
We have studied Ratio and Proportion. The statement, "the numbers a and b are in the ratio m/n " is also written as, "the numbers a and b are in proportion m:n". For this concept we consider positive real numbers. We know that the lengths of line segments and area of any figure are positive real numbers. We know the formula of area of a triangle.
Area of a triangle =1/2 Base × Height
Let’s find the ratio of areas of any two triangles.
Ex. In D ABC, AD is the height and BC is the base. In D PQR, PS is the height and QR is the base.
`A(triangle ABC)/A(trianglePQR)=(1/2xxBCxxAD)/(1/2xxQRxxPS)`
`A(triangle ABC)/A(triangle PQR)=(BCxxAD)/(QRxxPS)`
Hence the ratio of the areas of two triangles is equal to the ratio of the products of their bases and corrosponding heights. Base of a triangle is b1 and height is h1. Base of another triangle is b2 and height is h2 .
Then the ratio of their areas =`(b_1xxh_1)/(b_2xxh_2)`
Suppose some conditions are imposed on these two triangles,
Condition 1: If the heights of both triangles are equal then :
`A(triangle ABC)/A(triangle PQR)=(BCxxh)/(QRxxh)="BC"/QR"`
`A(triangle ABC)/A(triangle PQR)=b_1/b_2`
Property: The ratio of the areas of two triangles with equal heights is equal to the ratio of their corresponding bases.
Condition 2: If the bases of both triangles are equal then -
`A(triangle ABC)/A(triangle APB)=(ABxxh_1)/(ABxxh_2)`
`A(triangle ABC)/A(triangle APB)=h_1/h_2`
Property: The ratio of the areas of two triangles with equal bases is equal to the ratio of their corresponding heights.
