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Hi All,
I have posted a video on YouTube to discuss about
Circles : Basics and PropertiesAttached 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 CircumferenceArea of a Circle with radius r, A = ∏ \(r^2\)Circumference of a Circle with radius r, C = 2 ∏ rCentral Angle = 360˚Semicircle : Area and CircumferenceArea of a Semicircle with radius r, A = \(\frac{∏r^2}{2}\)Circumference of a Semicircle with radius r, C = ∏ r + 2rCentral Angle = 180˚Arc of a CircleArc 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 CircleSector 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 CirclesPROP 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 circlePROP 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 QuadrilateralA 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.Hope it Helps!
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