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Circle Properties Part I
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A circle is a set of all points in a plane that are the same distance from a fixed point in a plane The set of points form the. Circumference
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The line joining the centre of a circle and a point on the circumference is called the………………. Radius
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A is a straight line segment joining two points on the circle chord
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A chord that passes through the centre is a ………………………. diameter
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A……………………… is a straight line that cuts the circle in two points secant
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An arc is part of the circumference of a circle Major arc Minor arc
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A ……………………is part of the circle bounded by two radii and an arc sector Minor sector major sector
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A ……………………is part of the circle bounded by a chord and an arc segment Minor segment major segment
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The arc AB subtends an angle of at the centre of the circle. A B O Subtends means “to extend under” or “ to be opposite to”
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Instructions: Draw a circle Draw two chords of equal length Measure angles AOB and DOC A B C D O What do you notice?
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Equal chords subtend equal angles at the centre
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Conversely Equal angles at the centre of a circle stand on equal arcs
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Instructions: select an arc AB subtend the arc AB to the centre O subtend the arc AB to a point C on the circumference Measure angles AOB and ACB B O A C What do you notice?
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Instructions: select an arc AB subtend the arc AB to the centre O subtend the arc AB to a point C on the circumference Measure angles AOB and ACB B O A C What do you notice?
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22 The angle that an arc of a circle subtends at the centre is twice the angle it subtends at the circumference
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Instructions: select an arc AB select two points C, D on the circumference subtend the arc AB to a point C on the circumference subtend the arc AB to a point D on the circumference Measure angles ACB and ADB B O A C D
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Instructions: select an arc AB select two points C, D on the circumference subtend the arc AB to a point C on the circumference subtend the arc AB to a point D on the circumference Measure angles ACB and ADB B O A C D What do you notice?
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Angles subtended at the circumference by the same arc are equal
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Instructions: Draw a circle and its diameter subtend the diameter to a point on the circumference Measure ACB C B What do you notice? A
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An angle in a semicircle is a right angle
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γ Instructions: Draw a cyclic quadrilateral (the vertices of the quadrilateral lie on the circumference Measure all four angles β What do you notice?
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180- The opposite angles of a cyclic quadrilateral are supplementary 180-
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180- If the opposite angles of a quadrilateral are supplementary the quadrilateral is cyclic
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β Instructions: Draw a cyclic quadrilateral Produce a side of the quadrilateral Measure angles and β
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If a side of a cyclic quadrilateral is produced, the exterior angle is equal to the interior opposite angle
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Circle Properties Part IItangent properties
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A tangent to a circle is a straight line that touches the circle in one point only
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Tangent to a circle is perpendicular to the radius drawn from the point of contact.
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Tangents to a circle from an exterior point are equal
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When two circles touch, the line through their centres passes through their point of contact Point of contact External Contact
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When two circles touch, the line through their centres passes through their point of contact Point of contact Internal Contact
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The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment
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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 A B BA 2 =BC.BD C D B=external point
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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 A B BA 2 =BC.BD C D Note: B is the crucial point in the formula
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Circle Properties Chord properties
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A B C D X AX.XB=CX.XD Triangle AXD is similar to triangle CXB hence
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A B C D X AX.XB=CX.XD Note: X is the crucial point in the formula
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Chord AB and CD intersect at X Prove AX.XB=CX.XD A B C D X In AXD and CXB AXD = CXB (Vertically Opposite Angles) DAX = BCX (Angles standing on same arc) ADX = CBX (Angles standing on same arc) AXD CXB Hence (Equiangular ) AAA test for similar triangles
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A B C A perpendicular line from the centre off a circle to a chord bisects the chord
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A B C Conversley: A line from the centre of a circle that bisects a chord is perpendicular to the chord
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A B C Equal chords are equidistant from the centre of the circle
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A B C Conversley: Chords that are equidistant from the centre are equal
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Quick Quiz
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a 40 a= 40
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b b=b= 80 C
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d 60 d=d= 120 C
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f 55 f=f= C
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m=m= 62 C 62 mm
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e e=e= 90 C
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x=x= 12 C 102 12 cm x cm
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k 70 k=k= 35 C
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a 120 a= 50 10
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x 100 x=x= 50 C
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y y=y= 55 C 35
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Quick Quiz
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answer= A 105 75 Which quadrilateral is concyclic? A B C 100 110 20 140
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c 60 c = 60 C Tangent
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g g=g= 90 C Tangent
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h=h= 4 C 4cm h cm Tangent
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m 40 m = 50 C Tangent y = 50 y
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a=a= 65 C 50 Q a P R PQ, RQ are tangents
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n=n= 5 C 10 4 8 n nx8=4x10 8n =40 n =5
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q=q= 25 C 10 4 q 4q=10 2 4q=100 q=25
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x=x= 12 C 8 4 x 4(4+x)=8 2 4(4+x)=64 4+x=16 x=12 BA 2 =BC.BD
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k=k= 5 C 8m k 3m K 2 =3 2 +4 2 K =5
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