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I have posted a video on YouTube to discuss about Circles : Basics and Properties
Attached pdf of this Article as SPOILER at the top! Happy learning!
Following is Covered in the Video
Theory
¤ What is a Circle? ¤ Circle Geometry Definitions ¤ Circle : Area and Circumference ¤ Semicircle : Area and Circumference ¤ Arc of a Circle ¤ Sector of Circle ¤ Properties of Circles
What is a Circle?
A Circle is a 2D figure which is formed by joining all the points in a 2D plane which are at a fixed distance (i.e radius) from a single point. (i.e center of the circle)
Circle Geometry Definitions
¤ Radius – A line segment joining the center of the circle to any point on the circle. (Ex: OA) ¤ Chord– A line segment whose two end points lie on the circle (Ex: BC) ¤ Diameter– A chord which passed through the center. (Ex: DE) ( Diameter = 2* Radius )
¤ Secant– A line which cuts the circle at two points. (Ex: line s) ¤ Tangent– A line which touches circle at only one point. (Ex: line t)
Circle : Area and Circumference
Area of a Circle with radius r, A = ∏ \(r^2\)
Circumference of a Circle with radius r, C = 2 ∏ r
Central Angle = 360˚
Semicircle : Area and Circumference
Area of a Semicircle with radius r, A = \(\frac{∏r^2}{2}\)
Circumference of a Semicircle with radius r, C = ∏ r + 2r
Central Angle = 180˚
Arc of a Circle
Arc of a circle is a part of the Circumference of the circle.
Length of Arc AB, which subtends angle Θ at the center,
AB = \(\frac{𝛩}{360˚}\)∗ 2 ∏ r
Sector of Circle
Sector of a circle is a part of the circle made by the arc of the circle and the two radii connecting the arc to the center of the circle.
Area of sector OACB, which subtends angle Θ at the center, Area of OACB =\( \frac{𝛩}{360˚}∗ ∏ r^2\)
Circumference of Sector OACB is given by Circumference of OACB = \(\frac{𝛩}{360˚}\)∗ 2 ∏ r + 2r
Properties of Circles
PROP 1: A Chord subtends same angle at any point on the circle.
PROP 2: Angle subtended by the chord at the center is twice the angle subtended by the chord at any other point on the circle.
PROP 3: Diameter subtends 90˚ at any point on the circle
PROP 4: From an external point there are only two tangents which can be drawn to a circle and the length of these tangents is equal.
PROP 5: A tangent always makes 90˚with the line joining the point of tangency (point of intersection of the tangent with the circle) to the center of the circle.
PROP 6: Cyclic Quadrilateral A quadrilateral whose all 4 vertices lie on the circumference of the circle is called a Cyclic Quadrilateral.
Sum of all the angles of Cyclic Quadrilateral = 360 ˚ ∠ A + ∠ B + ∠ C + ∠ D = 360 ˚
Sum of diagonally opposite angles = 180 ˚ ∠ A + ∠ C = 180 ˚ ∠ B + ∠ D = 180 ˚
PROP 7: Perpendicular drawn from the center of the circle to a chord bisects the chord.
PROP 8: Equal chords are equidistant from the center. Or
Chords which are equidistant from the center are equal.
PROP 9: Equal chords subtend same angle at the center of the circle. Or
Chords which subtend same angle at the center of the circle are equal.