Rayat shikshan sanstha’s Kanya vidyalaya pimpri – waghere pune – 17 th Teachers name- Mrs.Manisha Vikas Nikam M.sc. B.Ed.

UNIT - CIRCLE

CIRCLES OLYMPIC

WINDOW

Circle 1.Tangent and its Properties . 2.Number of tangents to a circle through a given point. 3.Touching circles. 4.Arc of a circle. 5.Angle subtended by an arc. 6.Cyclic quadrilateral and its properties. 7.Tangent-Secant Theorem. 8.Properties of intersecting Secants

TANGENT AND ITS PROPERTIES x y z A line in the plane of a circle which intersects the circle in one and only one point is called tangent to the circle and the point of intersection is called the point of contact. m l .O P

Theorem 2.1-A tangent at any point of a circle is perpendicular to the radius through the point of contact. Given-Line I is the tangent to the circle with centre O at the point of contact P. To prove-Line l seg OP. l O . P l O . Q B P

Theorem 2.2-The line perpendicular to a radius of a circle at its outer end is tangent to a circle. Given-A circle with centre O ; radius OP, where P is a point on the circle; line l, which is perpendicular to the radius OP at the point P. To prove- The line l is tangent to the circle. O P Q

Number of tangents to a circle through a given point. Theorem 2 Number of tangents to a circle through a given point. Theorem 2.3-The length of the two tangent segments to a circle drown from an external point are equal. Given- A circle with centre O, an external point P of a circle and two tangents through the point P are touching the circle at the point A and B. To prove – PA = PB A O P B

Theorem 2.2-The line perpendicular to a radius of a circle at its outer end is tangent to a circle. Given-A circle with centre O ; radius OP, where P is a point on the circle; line l, which is perpendicular to the radius OP at the point P. To prove- The line l is tangent to the circle. O P Q

Touching Circles Given two circles in a plane are said to be touching circles or tangent circles if they have one and only one common point. Theorem 2.4-If two circles are touching circles then the common point lies on the line joining their centre. A R R P Q P Q B A P Q A P Q Given-Circles with centers P and Q are Touching in point A which is the common Point of two circles. To prove- The point A lies on Line PQ

Arc of a circle P P B P O A O C B A A B Q Q Q <ABC is inscribed in The arc APC or arc ABC Arc AQC is intercepted by <ABC Major arc- arc APB Minor arc- arc AQB Central angle- < AOB m ( arc AQB ) = m < AOB m ( arc AQB )+ m ( arc AQB ) =360°

Angle subtended by an arc The angle subtended by the chord AB at the point C on the circle is < ACB This angle is Called the angle subtended by the arc APB at the point C on the circle. C O B A P The central angle < AOB is called the angle subtended by the chord AB or angle subtended by the arc APB.

Inscribed angle theorem - The measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle subtended by the same arc at the centre. A O C Given – In a circle with centre O,< BAC is Subtended by the minor arc BPC. D B P To prove - < BAC = ½ < BOC . This theorem can also be stated as follows: - The measure of an angle subtended by an arc at a point on the circle is half of the measure of the arc. In fig. < BAC = ½ m(arc BPC)

Inscribed angle theorem –Corollary. Corollary – 1- An angle subtended by a semicircle at a point on the circle is a right angle. In fig. < BAC = 90° A C O B P A2 Corollary -2-Angle subtended by the same arc are congruent. In fig. < BAC = < BA1C = < BA2C = C A A1 O B P

Cyclic Quadrilateral and its properties. Theorem- The opposite angles angles of a cyclic quadrilatral are supplimentary A B Given - ABCD is cyclic quadrilateral. To prove- < BAD + < BCD = 180° and < ABC + < ADC = 180° O D C

Cyclic Quadrilateral and its properties. Converse – If a pair of opposite angles of a quadrilateral is Supplementary , then the quadrilateral is cyclic. E D C Given – A quadrilateral ABCD in which < A + < C = 180° To prove - ABCD is cyclic quadrilateral. B A D E C B A

Cyclic Quadrilateral and its properties. Corollary – An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle. Given - ABCD is a cyclic quadrilateral. Side DC OF A cyclic quadrilateral is produced and point E IS ON it. To prove - < BCE = < BAD. A B D C E

TANGENT SECANT THEOREM –If an angle with its vertex on the circle whose one side touches the circle and the other intersects the circle in two points , then the measure of the angle is half the measure of its intercepted arc. A A A F O O F O E E E F D B D B C B C C D (2) (3) (1) Given - Let < ABC be an angle where vertex B lies on a circle with centre O, line BC is a tangent at B and line BA is a secant of the circle intersecting the circle at A. Point E and F are points of the circle such that E is in the interior and F is in the exterior of the circle. To prove - < ABC = ½ m( arc AEB )

Angles in Alternate segments Angles in Alternate segments. Theorem – If a secant is drown through the point of contact of a tangent to a circle then the angle which this tangent makes with the chord contained in the secant are equal respectively to the angles subtended by the chord in the corresponding alternate segments. A Given – Let AB be a secant of a circle , A,B be the point On the circle and line DC be the tangent to the circle at B. Let E and F be any two points on the circle such that they Are in the alternate segments R1 and R2 respectively. E R1 F R2 D B C To prove - < AEB = < ABC and < ABD = < AFB.

Converse – If a line drown intersecting to a secant of a circle at a common point of the secant and the circle so that the angle formed by it with the chord contained in its secant is equal to an angle subtended by the chord in the alternate segment, then that line is tangent to a circle. RR Given - Seg AB is a secant of a circle, point A , B are on the circle and line PQ is such that P-A-Q and < BAQ = < ACB where the point C is in corresponding alternate segment. C B R1 R2 Q1 P A Q P1 To prove – Line PQ is tangent at A.

Properties of intersecting secants Properties of intersecting secants. Theorem – If two secants of a circle intersects inside or outside the circle then the area of the rectangle formed by the two line segments corresponding to one secant is equal in area to the rectangle formed by the two line segments corresponding to the other. A D PP A P B P P B C D C Given – Secants AB and CD intersects in point P To prove – (Area of rectangle formed = (Area of rectangle formed by side AP and BP) by side CP and DP) i.e. AP BP =CP PD

Properties of intersecting secants Theorem – If a secant and a tangent of a circle intersect in a point out side the circle then the area of rectangle formed by the two line segments corresponding to the secant is equal to the area of the square formed by the line segment corresponding to the tangent. T Given- A secant through the point P intersect the circle in point A and B. Tangent drown through the point P touches the circle in point T. P A B To prove – PA PB = PT²

Questions- One mark 1. From fig. write the names of l m 1. From fig. write the names of tangent and secant. 2.How many tangents to a circle can be drown through an external point of the circle? 3.Radii of two internally touching circles are 7 cm and 4 cm . Find the distance between their centre. 4.What is the measure of an angle subtended by a semi circle at a point on the circle? 5. From fig. write a pair of congruent angles. 6. From fig. write a pair of supplementary angles. A D E C B

Questions- Two marks 1]If diameters of two externally touching circles are 11 cm and 17 cm. Find the distance between their centre. 2] If m (arc PSR) = 70°. Find m < PQR. 3] In fig. circle is inscribed in ABC. If AR = 5, BQ = 7. Find AB. 4] In fig. DEFG is a cyclic quadrilateral. <DGF = 110°,find <DEF and <FGH. 5] In fig. line DG is tangent to the circle, seg RD is a chord.If m(arc RAD) = 120°, find <RDG. P S A R Q C R P Q B D G H E F R A D G

Questions- Three marks B 1) In fig . P is the centre of the circle and AB and AC are tangent segments to the circle. If m< BAC = 35°, find < BPC. 2) Explain what is wrong with the information marked on the figure. 3)Chords AB and CD of a circle intersect at P, and AP = 6, PB = 4, CP = 8. Find PD. 4) Two circles touch each other internally. From a point T on tangent at P, tangents TA and TB are drown to the circle . A and B are points of contact . Prove TA = TB . A OO P C 160° PP R P 70° Q T A B P

Questions- Four marks 1)Circles with centre P,Q,R touch externally at A,B,C PQ = 18 , QR = 13, PR = 15.Find their radii . 2)Incircle of ABC touch sides AB,BC,CA respectively in D,E and F. AB = 13,BC = 12,AC = 9 . Find AB, BC, CF. 3) Two tangents TP and TQ are drown to a circle with centre O from an external point T. Prove that < PTQ = 2 < OPQ. 4) In cyclic ABCD , AB CD . Prove that AB = BC. P T O Q

Questions- Five marks 1)If in cyclic ABCD , AD = BC , prove AB DC. 2) A quadrilateral ABCD is drown to circumscribed a circle. Prove that AB + CD = AD + BC . M is mid point of seg AB . Semi circles are drown on seg AM,MB , and AB as diameter . Circle with centre O touches all the three semi circles Prove that r = 1/6 AB . 4)Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. R D C Q S B P A R r O Q P D B C M A

UNITWISE DISTRIBUTION OF MARKS MARKS WITH OPTION 1 Similarity 12 2 Circle 10 3 Geometric Construction 4 Trigonometry 5 Co-ordinate Geometry 8 6 Mensuration 60