Chord of Circle Theorems
Theorem 1: Perpendicular line drawn from the centre of a circle to a chord bisects the chord.
Given:
Chord AB and line segment OC is perpendicular to AB
To prove:
AC = BC
Construction:
Join radius OA and OB
Proof:
In ΔOAC and ΔOBC
∠OCA = ∠OCB (OC is perpendicular to AB)
OA = OB (Radii of the same circle)
OC = OC (Common Side)
So, by RHS congruence criterion ΔOAC ≅ ΔOBC
Thus, AC = CB (By CPCT)
Converse of the above theorem is also true.
Theorem 2: Line drawn through the centre of the circle to bisect a chord is perpendicular to the chord.
(For reference, see the Image used above.)
Given:
C is the midpoint of the chord AB of the circle with the centre of the circle at O
To prove:
OC is perpendicular to AB
Construction:
Join radii OA and OB also join OC
Proof:
In ∆OAC and ∆OBC
AC = BC (Given)
OA = OB (Radii of the same circle)
OC = OC (Common)
By SSS congruency criterion ∆OAC ≅ ∆OBC
∠1 = ∠2 (By CPCT)…(1)
∠1 + ∠2 = 180° (Linear pair angles)…(2)
Solving eq(1) and (2)
∠1 = ∠2 = 90°
Thus, OC is perpendicular to AB.
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Radius of Circle
Radius of Circle: The radius of a circle is the distance from the circle’s center to any point on its circumference. It is commonly represented by ‘R’ or ‘r’. The radius is crucial in nearly all circle-related formulas, as the area and circumference of a circle are also calculated using the radius.
In this article, we are going to learn about the Radius of the Circle in detail, including its Formula, Equation, and How to Find it with the help of Examples.
Table of Content
- What is the Radius of Circle?
- Radius of a Circle Definition
- Diameter of Circle
- Radius, Diameter and Chord
- Secant to Circle
- Tangent to Circle
- Non-Intersecting Lines
- Radius Formula
- How to Find Radius of Circle?
- Radius of Sphere
- Radius of Circle Equation
- Chord of Circle Theorems
- Radius of Circle Examples
- Practice Questions on Radius of Circle