Circle Properties PDF

CIRCLE Presented by: Liezelle Mendoza  EXPLORING CIRCUMFERENCE, ARC LENGTH, AND ANGLE PROPERTIES Objectives : At the end of the discussion you will be able to: - Describe the relationship between the radius, diameter, and circumference of a circle. - Identif...

CIRCLE Presented by: Liezelle Mendoza  EXPLORING CIRCUMFERENCE, ARC LENGTH, AND ANGLE PROPERTIES Objectives : At the end of the discussion you will be able to: - Describe the relationship between the radius, diameter, and circumference of a circle. - Identify different types of angles within a circle (e.g., central, inscribed, tangent). - Apply circle properties to solve geometric problems.  What is a Circle? A circle is a closed shape formed by all points that are the same distance from a central point. It is named by its center. What is a Radius? It is a line segment from the center to any point of the circle.  What is a Chord? It is a line segment whose endpoints lie on the circle. What is a Diameter? It is a chord containing the center. ACTIVITY Name the ff. 1. Circle= 2. Radii= 3. Diameter= 4. Chord= Circumference and Arc Length Measuring the Outside  What is a Circumference? The circumference of a circle is the distance around the circle. It is another name for the perimeter of a circle.  Formula for Circumference C=2πr C= circumference r= radius C=πd d= diameter π= 3.14 Example: C=2πr C=2x3.14x4cm C=25.12cm C=πd C=3.14x8cm C=25.12cm What is an Arc Length? Arc length is better defined as the distance along the part of the circumference of any circle or any curve (arc).  Formula for Arc Length S=θr (where θ is in radian) S=(θ/360)2πr (where θ is in degrees)  What is a Sector Area? The area of sector of a circle is the space enclosed within the boundary of the sector. A sector always originates from the center of the circle.  Formula for Sector Area A=1/2θr² (where θ is in radian) A=(θ/360)(πr²) (where θ is in degrees) Arc and Angle Properties Angles in Circles  What is an Arc? An arc of a circle is two points on the circle and the continuous part of the circle between the two points. The two points are called the endpoints of the arc.  Arcs of a Circle A semicircle is an arc of a circle whose endpoints are the points of the diameter. ⌒ SML ⌒ SEL ⌒ SIL  Arcs of a Circle A minor arc is an arc of a circle that is smaller than a semicircle. ⌒ SM ⌒ MI ⌒ IL  Arcs of a Circle A major arc is an arc of a circle that is larger than a semicircle. ⌒ SME ⌒ SEI ⌒ ILS  Angles of a Circle A central angle is an angle whose vertex is the center of the circle. ∠ S N M m⌒ SM= 50  Angles of a Circle A inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. ∠ I S L m⌒ IL= 60 ACTIVITY: LET’S TRY ⌒O = 1. mM 2.m⌒ OL= 3. m⌒ ML= 4. m S⌒ ML= 5. m S⌒ OI=  Secant and Tangent of a Circle A secant of a circle is a line that intersects the circle in two points. ↔ Secant: ML  Secant and Tangent of a Circle A tangent of a circle is a line lying on the same plane that intersects the circle in excatly one point. Tangent: line l Point of tangency: E  Angle Properties Circle Theorem 1: The alternate segment The angle that lies between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.  Angle Properties Circle Theorem 2: Angles at the centre and at the circumference The angle at the centre is twice the angle at the circumference.  Angle Properties Circle Theorem 3: Angles in the same segment Angles in the same segment are equal.  Angle Properties Circle Theorem 4: Angles in a semicircle The angle in a semicircle is 90 degrees. Circle Theorem 5: Chord of a Angle Properties circle The perpendicular from the centre of a circle to a chord bisects the chord (splits the chord into two equal parts). Circle Theorem 6: Tangent of Angle Properties a circle The angle between a tangent and radius is 90 degrees. Tangents which meet at the same point are equal in length.  Circle Theorem 7: Angle Properties Cyclic quadrilateral The opposite angles in a cyclic quadrilateral total 180°. KEY TERMINOLOGY: Center: The central point Radius: The distance from the center to any point on the circle. Chord: A line segment connecting two points on the circle Diameter: The distance across the circle passing through the center. Circumference: The distance around the circle. pi: (or π) The ratio of the circumference of a circle to its diameter. ACTIVITY: identify each of the following as related to the specified circle. ACTIVITY: Answer the following. ACTIVITY: Answer the following. Great Job!