Circle Theorems for the Tangents
The above discussion was important to understand the theorems related to circles. Only two theorems have been added to the syllabus, which defines the properties of tangents.
Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Note: The point of contact is the intersection point, where radius, and tangent of a circle meet.
Explanation:
A circle has infinitely many lines of symmetry. Let us consider a line, passing the centre of the circle. It cuts the circle into halves, and at the point of contact, a tangent is formed. Due to the symmetry, there is no reason to assume that a > b or b > a. Hence, radius and tangent form right angles.
Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
Explanation:
An external Point A cuts the circle at two points, B, and C. Line AB, and AC, are tangents to the circle. From theorem 1, as radius, and tangent intersect at right angles, make line OB, and OC, where ∠ OBA = ∠ OCA = 90°.
By Concurrency of Triangles:
In △ABO and △ACO,
⇒ AO = AO (Common)
⇒ OB = OC (Radius)
⇒ ∠ABO = ∠ACO (right-angle each)
By RHS, △ABO ≅ △ACO (Congruence Triangle)
Hence, AB = AC (Length of tangent is equal)
Note: ∠BOA = ∠AOC, as above triangles are congruent.
Also, Read
Circles Class 10 Maths Notes Chapter 10
CBSE Class 10 Maths Notes Chapter 10 Circles are an excellent resource, for knowing a particular chapter’s concepts in a crisp, friendly manner. Our articles help to learn children in their language, with proper images and solved examples for a better understanding of the concepts.
Chapter 10 of the NCERT Class 10 Maths textbook delves into the world of Circles, and their tangent theorems. It covers various topics such as introduction to tangents, properties of tangents, and theorems for circles on tangents These notes are designed to give students a comprehensive summary of the entire chapter and include all the essential topics, formulae, and concepts needed to succeed in their exams.