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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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