Circle Geometry. Properties of a Circle Circle Theorems:! Angles and chords! Angles! Chords! Tangents! Cyclic Quadrilaterals

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1 ircle Geometry Properties of a ircle ircle Theorems:! ngles and chords! ngles! hords! Tangents! yclic Quadrilaterals 1

2 Properties of a ircle Radius Major Segment iameter hord Minor Segment Tangent oncyclic points form a yclic Quadrilateral Sector rc Tangents Externally and Internally oncentric circles 2

3 ircle Theorems l! Equal arcs subtend equal angels at the centre of the circle.! If two arcs subtend equal angles at the centre of the circle, then the arcs are equal. l = r l! Equal chords subtend equal angles at the centre of the circle.! Equal angles subtended at the centre of the circle cut off equal chords. S = (radius of circle) = (vert. opp. ngles) S = (radius of circle) (SS) = (corresponding sides in ' s ) 3

4 ! perpendicular line from the centre of a circle to a chord bisects the chord.! line from the centre of a circle that bisects a chord is perpendicular to the chord. R M = M (straight line) H = (radius of circle) S M = M (common) M M (RHS) M = M (corresponding sides in ' s ) M 4

5 ! Equal chords are equidistant from the centre of the circle.! hords that are equidistant from the centre are equal. N M R N = M = 90 ( line from the centre of a circle that bisects a chord is perpendicular to the chord) H = (Radius of ircle) S N = M (given) N M (RHS) 5

6 Internally! The products of intercepts of intersecting chords are equal X.X = X.X X Prove X X X X = X (vertically opp) X = X (ngle standing on the same arc) X = X (ngle sum of triangle) orrespond sides X X = X X X. X = X. X 6

7 Externally! The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point. (X) 2 = X.X X Externally Prove X X X = X (common) X = X (ngle in alternate segment) X = X (ngle sum of triangle) orrespond sides X X = X X ( X ) 2 = X. X X 7

8 ! The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. 2 α β β Let = α Let = β = = (radius of circle) = α (base angles of isosceles ) = β (base angles of isosceles ) = 2α (exterior angle = two opposite interior angles) = 2β (exterior angle = two opposite interior angles) α = α + β = 2( a + β )! ngle in a semicircle is a right angle

9 ! ngles standing on the same arc are equal. Prove X X X X = X (vertically opp) X = X (ngle standing on the same arc) X = X (ngle sum of triangle) orresponding angles of similar triangles are equal 9

10 ! Tangents to a circle from an exterior point are equal. Prove R = (90 ) H = (common) S = ) (radii) RHS = (corresponding sides in congruent triangles) 10

11 ! When two circles touch, the line through their centres passes through their point of contact. 11

12 ! The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. X Let X = α Let Q = β α + β = 90 X = β (angle in semicircle is 90, complementary angle) = X (angle on the same arc) = Q Q 12

13 α! The opposite angles in a cyclic quadrilateral are supplementary.! If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. 180 β β 180 α α! The exterior angle of s cyclic quadrilateral is equal to the interior opposite angle. α 13

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