MATHS PROJECT

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MATHS PROJECT

• CIRCLES
• BY
• L . SRUTHI
• 10TH A

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INTRODUCTION

• A circle is collection of all points in a plane
  which are at a constant distance (radius) from a
  fixed point (centre).
• There are also various terms related to a circle
  like a chord ,segment ,sector ,arc etc..
• So, let us consider a circle a line PQ. there
  can be three possibilities like

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1.In fig (1) line PQ is called a non-intersecting
line2.In fig (2) line PQ is called a secant line
to its circle 3.In fig (3) line PQ is called a
tangent to its circle.

fig 1 fig 2 fig 3

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Tangent to a circle

• A tangent to a circle is a line
• that intersects the
• Circle at only one point

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More to know

• Tangent comes from latin word tangere, which
  means to touch was introduced by the Danish
  mathematician Thomas Fineke in 1583

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THEOREM 10.1

• The tangent at any point of a circle is
  perpendicular to the radius through the point f
  contact.

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Proof

• given A circle with centre O a tangent XY to
  the circle at pint P.
• TO PROVE OP is perpendicular to XY.
• Construction take a point Q on XY other than P
  join OQ.
• PROOF the point Q must lie outside the circle
  as XY is a tangent. Therefore
• OQ is longer than the radius OP of the circle.
  i.e.
• OQ gt OP
  
  

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Since this happens for every point P.OP is the
line XY except the point O to the points of XY.
So OP is perpendicular to XY.
o
X P Q Y
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REMARKS

• By this theorem , we can also conclude that any
  point on a circle there can be only one tangent.
• The line containing the radius through the point
  of contact is also sometimes called the normal
  to the circle at the point.

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EXERCISE 10.1

• How many tangents can a circle have?
• Fill in the blanks
  
• a) a tangent to a circle intersects it in ____
  point (s).
• b) A line intersecting a circle in two points is
  called a ______ .
• c) A circle can have _____ parallel tangents at
  most .
• d) The common point of a tangent to a circle
  the circle is called ________________ .
• 
  

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EXERCISE 10.1

• 3) A tangent PQ at a point P of a circle of
  radius 5 cm meets a line through the centre O at
  a point Q so that OQ 12 cm . Length PQ is
• a)12 cm b)13 cm c)8.5 cm
• 4) Draw a circle two lines parallel to a given
  line such that one is a tangent other, a secant
  to the circle.

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NUMBER OF TANGENTS FROM A POINT ON A CIRCLE

• There are three cases
• There is no tangent to a circle passing through a
  point lying inside the circle.
• There is one only one tangent to a circle
  passing through a point lying on the circle.
• There are exactly two tangents to a circle
  through a point ling outside the circle.

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Theorem10.2

• Given A circle with centre O , a point P lying
  outside the circle two tangents PQ , PR on the
  circle from P .
• To prove PQPR
• CONSTRUCTION Join OP,OR OQ .

Q
P O
R
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Proof

• Then OQP ORP are right angles , because
  these are angles between the radii tangents ,
  according to theorem 10.1 they are right angles .
• Now in rt s OQP ORP
• OQOR (radii of the circle)
• OPOP (common)
• OQP ORP ( RHS)
• Therefore ,PQPR (CPCT)

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EXAMPLE 1

• Q Prove that in two concentric circles , the
  chord of the larger circle , which touches the
  smaller circle , is bisected at the point of
  contact.
• Sol GIVEN Two concentric circles with centre O
  chord AB of the circle larger circle which
  touches the smaller circle at point P.
• TO PROVE APBP
• CONSTRUCTION Join AP

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Proof

• Then AB is a tangent to smaller circle at P OP
  is its radius.
• Therefore, by theorem 10.1
• OP AB
• Now AB is a chord of the larger circle OP AB
• Therefore, OP is the bisector of the chord AB ,
  as the perpendicular from the centre bisects the
  chord

o
A P B
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EXAMPLE 2

• Q. Two tangents TP TQ are drawn to a circle
  with centre O from an external point T. Prove
  that
• PTQ 2 OPQ

P
T O
Q
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EXAMPLE 3

• Q. PQ is a chord of length 8 cm of a circle of
  radius 5 cm. The tangents at P Q intersect at a
  point T. Find the length TP.

P
T R O
Q
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EXERCISE 10.2

• In Q.1 to 3, choose the correct option justify
• From a point Q, the length of the tangent to a
  circle is 24 cm the distance of Q from the
  centre is 25 cm. The radius of the circle is
• 7 cm
• 12 cm
• 15 cm
• 24.5 cm
• 2) In fig, if TP TQ are two tangents to a
  circle with centre O so that POQ 110, then
  PTQ is equal to
• 60
• 70
• 80
• 90

P T
O Q
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EXERCISE 10.2

• 3)If tangents PA PB from a point P toa circle
  with centre O are inclined to each other at angle
  of 80, then POA is
• 50
• 60
• 70
• 80
• 4)Prove that the tangents drawn at the ends of a
  diameter of a circle are parallel.
• 5)Prove that the perpendicular at the point of
  contact to the tangent to a circle passes through
  the centre.
• 6)The length of a tangent from a point A at
  distance 5 cm from the centre of the circle is 4
  cm. find the radius of the circle.

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EXERCISE 10.2

• 7)Two concentric circles are of radii 5 cm 3 cm
  . Find the length of the chord of the larger
  circle which touches the smaller circle.
• 8)A quadrilateral ABCD is drawn to circumscribe a
  circle . Prove that ABCD ADBC

D R C
S Q
A P B
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EXERCISE 10.2

• 9)In fig, XY XY are two parallel tangents to
  a circle with centre O another tangent AB with
  point of contact C intersecting XY at A XY at
  B. prove that AOB90.
• 10)Prove that the angle between the two tangents
  drawn from an external point to a circle is
  supplementary to the angle subtended by the line
  segment joining the points of contact at the
  centre.
• 11)Prove that the parallelogram circumscribing a
  circle is a rhombus.

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EXERCISE 10.2

• 12)A triangle ABC is drawn to circumscribe a
  circle of radius 4 cm such that the segments BD
  DC into which BC is divided by the point of
  contact D are of lengths 8 cm 6 cm respectively
  . Find the sides AB AC.

A
O
C B
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EXERCISE 10.2

• 13)Prove that opposite sides of a quad.
  Circumscribing a circle subtend supplementary
  angles at the centre of the circle.
• Summary
• 1.The meaning of tangent
• 2. The tangent to a circle is perpendicular to
  the radius through the point of contact .
• 3.The lengths of the two tangents from an
  external point to a circle are equal.

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Thank you