CLASS X_ : CHAPTER - 10 CIRCLES IMPORTANT FORMULAS & CONCEPTS Circle The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. > The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle. In the below figure, O is the centre and the length OP is the radius of the circle. P > The line segment joining the centre and any point on the circle is also called a radius of the circle. > A circle divides the plane on which it lies into three parts. They are: (i) inside the circle, which is, also called the interior of the circle; (ii) the circle and (iii) outside the circle, which is also called the exterior of the circle. The circle and its interior make up the circular region. > The chord is the line segment having its two end points lying on the circumference of the circle. > The chord, which passes through the centre of the circle, is called a diameter of the circle. > A diameter is the longest chord and all diameters have the same length, which is equal to two times the radius. > Apiece of a circle between two points is called an arc. Ww ‘The longer one is called the major are PQ and the shorter one is called the minor arc PQ. > The length of the complete circle is called its circumference. > The region between a chord and either of its arcs is called a segment of the circular region or simply a segment of the circle. There are two types of segments also, which are the major segment and the minor segment. v ‘The region between an are and the two radii, joining the centre to the end points of the arc is called a sector. The minor arc corresponds to the minor sector and the major arc corresponds to the major sector. v In the below figure, the region OPQ is the minor sector and remaining part of the circular region is the major sector. When two arcs are equal, that is, each is a semicircle, then both segments and both sectors become the same and each is known as a semicircular region. Scanned with CamScannerMajor sector Major segment PNG emeg 2 Points to Remember : > Acircle is a collection of all the points in a plane, which are equidistant from a fixed point in the plane > Equal chords of a circle (or of congruent circles) subtend equal angles at the centre. If the angles subtended by two chords of a circle (or of congruent circles) at the centre + (corresponding centre) are equal, the chords are equal The perpendicular from the centre of a circle to a chord bisects the chord. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. vv ‘There is one and only one circle passing through three non-collinear points. ¥ v Equal chords of a cirele (or of congruent circles) are equidistant from the centre (or corresponding centres). > Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal, > Iftwo ares of a circle are congruent, then their corresponding chords are equal and conversely, if two chords of a circle are equal, then their corresponding ares (minor, major) are congruent. > Congruent ares of a circle subtend equal angles at the centre. ‘The angle subtended by an arc at the centre is double the angle subtended by it at any point on v the remaining part of the circle. > Angles in the same segment of a circle are equal.\ v Angle in a semicircle is a right angle. v If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle. > The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. > If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. Secant to a Circle A secant to a circle isa line that intersects the circle at exactly two points. Tangent to a Circle A tangent to a circle is a line that intersects the circle at only one point. Scanned with CamScannerGiven two circles, there are lines that are tangents to both of them at the same time. © Ifthe circles are separate (do not intersect), there are four possible common tangents au © If the two circles touch at just one point, there are three possible tangent lines that are common to both: © If the two circles touch at just one point, with one inside the other, there is just one line that is a tangent to both: je. intersect at two points, there are two tangents that are common to both: CD © If the circles lie one inside the other, there are no tangents that are common to both. A tangent to the inner circle would be a secant of the outer circle. © If the circles overlap - Scanned with CamScanner© The tangent to a circle is perpendicular to the radius through the point of contact. © The lengths of tangents drawn from an external point to a circle are equal. & The centre lies on the bisector of the angle between the two tangents. ‘© “Ifa line in the plane of a circle perpendicular to the radius at its endpoint on the circle, then the line is tangent to the circle”, Scanned with CamScannerMCO WORKSHEET-I 1. Find the length of tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre. (a) 24cm (b) 27 cm (c) 26 cm (d) 25 cm 2. A point P is 26 cm away from the centre of a circle and the length of the tangent drawn from P to the circle is 24 cm, Find the radius of the circle (allem = (b)10em_— (©) 16em_—(d) 1Sem 3. Froman external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of the APCD. (a) 28. cm (b) 27cm (c) 26cm (d) 25cm A c A P 7? bs B B 4, In the above sided figure, PA and PB are tangents such that PA = 9cm and ZAPB = 60°. Find the length of the chord AB. (a)4em (b) 7 cm ()6cm (@9cm 5. In the below figure the circle touches all the sides. of a quadrilateral ABCD whose three sides are AB = 6 cm, BC = 7 cm, CD = 4 cm. Find AD. (a)4cm (b) 3. cm ()6cm (@) 9cm R c J Q s A Q AY 6. Inthe above sided Fig., if TP and TQ are the two tangents to a circle with centre O so that ZPOQ = 110°, then ZPTQ is equal to (a) 60° (b) 70° () 80° @ 90° 7. If tangents PA and PB from a point P to a circle with centre O are of 80°, then ZPOA is equal to (a) 60° (b) 70° (©) 80° (@) 50° ined to each other at angle Scanned with CamScanner8. 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. (a) 4em (b) 3. cm (©) 6cm (d) Scm 9, From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle. @4em — (b)7em_—()Gem_—() Sem 10. PT is tangent to a circle with centre O, OT = 56 cm, TP = 90 cm, find OP (a) 104cm = (b) 107cm_— (c) 106cm_—(d) 105 cm 11. TP and TQ are the two tangents to a circle with center O so that angle ZPOQ = 130°. Find 2PTQ. (a) 50° (b) 70° (©) 80° (@) none of these 12. From a point Q, the length of the tangent to a circle is 40 cm and the distance of Q from the centre is 41 cm, Find the radius of the circle. (a)4em (b) 3. cm (c) 6em (a) 9.cm 13. The common point of a tangent to a circle with the circle is called (a)centre —(b) point of contact (c) end point (d) none of these 14, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see below figure). Find the length TP. x (a) 2 cm (b) 2 em (c) Bem (d) none of these 15. The lengths of tangents drawn from an external point to a circle are equal. (a) half (b) one third (c) one fourth (d) equal Scanned with CamScannerMCO WORKSHEET-II CIRCLES 1. Inbelow Fig, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If Z DBC = 55° and ZBAC = 45°, find ZBCD. (a) 80° (b) 60° ©” (d) none of these D B ¢ ‘K ab/ A V B Cc D In above sided Fig, A,B and C are three points on a circle with centre O such that ZBOC = 30° and ZAOB = 60°, If D is a point on the circle other than the are ABC, find ZADC. (a) 45° (b) 60° (©) 90° (A) none of these 3. Achord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc (@) 150° (b) 30° (©) 60° (@) none of these 4. Acchord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the major are. (a) 150° (b) 30° (©) 60” (@) none of these 5. In the below Fig., ZABC = 69°, ZACB = 31°, find BDC. (a) 80° (b) 60° (c) 90° (d) 100° 6. Inthe above sided Fig., A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ZBEC = 130° and ZECD = 20°. Find BAC. (a) Lo (b) 150° (c) 90° (d) 100° 7. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ZDBC = 70°, ZBAC is 30°, find ZBCD. (a) 30° (b) 60° (90 (d) 100° Scanned with CamScanner8. ABCD is a cyclic quadrilateral. If ZBCD = 100°, ZABD is 30°, find ZABD. (a) 80° (b) 60° (©) 90° @” 9. ABCD is a cyclic quadrilateral. If ZDBC= 80°, ZBAC is 40°, find BCD. (a) 80° (b) 60° (©) 90" (@ 70° 10. ABCD is a cyclic quadrilateral in which BC is parallel to AD, ZADC = 110° and ZBAC = 50°. Find ZDAC (a) 80° (b) 60° (©) 90° (@ 170° 11. In the below figure, ZPOQ= 80°, find ZPAQ (a) 80° (b) 40" (©) 100° (d) none of these A c Q Pj a R P Q 12. In the above figure, ZPQR = 100°, where P, Q and R are points on a circle with centre O. Find ZOPR (a) 80° (b) 40° (©) 10° (d) none of these Scanned with CamScanner1 MCO WORKSHEET-III Distance of chord AB from the centre is 12 cm and length of the chord is 10 cm. Then diameter of the circle is A. 26cm B. 13cm Cc. V244 om D. 20cm ‘Two circles are drawn with side AB and AC of a triangle ABC as diameters. Circles intersect at a point D, Then A. ZADB and ZADC are equal B. ZADB and Z ADC are compementary C. Points B, D, C are collinear D. _ none of these The region between a chord and either of the arcs is called A. anare B. a sector C. segment D. asemicircle A circle divides the plane in which it lies, including circle in A. 2 parts B. 3 parts C. 4 parts D. 5 parts If diagonals of a cyclic quadrilateral are the diameters of a circle through the vertices of a quadrilateral, then quadrilateral is a A. parallelogram — B. square C. rectangle D. trapezium Given three non collinear points, then the number of circles which can be drawn through these three points are A. one B, zero Cc. two D. infinite In a circle with centre O, AB and CD are two diameters perpendicular to each other. The length of chord AC is A. 2AB B. V2 AB fos AB D. ae If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then A. ZAPB = ZAQB ZAPB + ZAQB = 180° ZAPB + ZAQB = 90° ZAPB + ZAQB = 180° pap Scanned with CamScanner9. Inthe above figure, ZPQR = 90°, where P, Q and R are points on a circle with centre O. Find reflexZPOR. (a) 180° (b) 140° (45° (@) none of these Q L> 10. In below Fig, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If Z DBC = 60° and ZBAC = 30°, find 2BCD. (a) 80° (b) 60° (©) 90° (@) none of these D Scanned with CamScannerPRACTICE QUESTIONS CLASS X: CHAPTER — 10 1. Prove that “The tangent at any point of a circle is perpendicular to the radius through the point of contact”. 2. Prove that “The lengths of tangents drawn from an external point to a circle are equal.” Prove that “The centre lies on the bisector of the angle between the two tangents drawn from an external point to a circle. 4, Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 em from the centre. 5. A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle. 6, From appoint Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre of the circle is 25 em. Find the radius of the circle. 7. The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle. 8. Prove that in two concentric circles, the chord of the bigger circle, which touches the smaller circle is bisected at the point of contact. 9. APQR circumscribes a circle of radius r such that angle Q = 90°, PQ = 3 cm and QR = 4 cm. Find r. le is a rhombus. 10. Prove that the parallelogram circumscribing a c OR fall the sides of a parallelogram touch the circle, show that the parallelogram is a rhombus. 11. ABC is an isosceles triangle in which AB = AC, circumscribed about a circle. Show that BC is bisected at the point of contact. 12, In Fig,, a circle is inscribed in a quadrilateral ABCD in which 2B = 90". If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius (r) of the circle. Scanned with CamScanner13. ABCD is a quadrilateral such that 2D =90°. A circle C(O, r) touches the sides AB, BC, CD and DA at P, Q, R and S respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm, find r. 14. An isosceles triangle ABC is inscribed in a circle. If AB = AC = 13 cm and BC = 10cm, find the radius of the circle. 15. Two tangents TP and TQ are drawn from a external point T to a circle with centre O, as shown in fig. If they are inclined to each other at an angle of 100° then what is the value of ZPOQ ? Q ? 16. The incircle of AABC touches the sides BC, CA and AB at D, E and F respectively. If AB = AC, prove that BD = CD. 17. XP and XQ are tangents from X to the circle with O, R is a point on the circle and a tangent through R intersect XP and XQ at A and B respectively. Prove that XA + AR = XB + BR. 18. A circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7em and CD = 4m. Find AD. Ss D Cc A - B e with centre O at P and Q respectively. PQ = Sem and radius of 19. TP and TQ are tangents to a circle is 5 cm. Find TP and TQ. 20. In the below figure PT is tangent to a circle with centre O, PT = 36 cm, AP = 24 cm, Find the radius of the circle. t Scanned with CamScanner21. In the below figure, find the actual length of sides of AOTP. 22. In the above sided figure, find the value of x. 23, Find the perimeter of DEFG. 24, Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ZPTQ = 2Z0PQ. 25. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. cle passes through the 26, Prove that the perpendicular at the point of contact to the tangent to a ci centre. 27. 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. 28. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 29. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC 30. 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. Scanned with CamScanner31. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. OR A circle touches all the four sides a quadrilateral ABCD. Prove that the angles subtended at the centre of the circle by the opposite sides are supplementary. 32. PA and PB are the two tangents to a circle with centre O in which OP is equal to the diameter of the circle. Prove that APB is an equilateral triangle, 33. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the center of the circle. 34. If PQ and RS are two parallel tangents to a circle with centre © and another tangent X, with point of contact C intersects PQ at A and RS at B. Prove that ZAOB = 90°. 35. The incircle of AABC touches the sides BC, CA and AB at D, E and F respectively. If AB = AC, prove that BD = DC. 36. Two tangents PA and PB are drawn to the circle with center O, such that APB = 120°. Prove that OP = 2AP. 37. A circle is touching the side BC of AABC at P and is touching AB and AC when produced at Q and R respectively. Prove that AQ= ¥ (Perimeter of A ABC) 38. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. A AD 39. In figure, chords AB and CD of the circle intersect at 0. OA = Sem, OB = 3cm and OC = 2.5em. Find OD. Scanned with CamScanner40. In figure. Chords AB and CD intersect at P. If AB = Sem, PB = 3em and PD = dem, Find the length of CD. 41. In the figure, ABC is an isosceles triangle in which AB = AC. A circle through B touches the side AC at D and intersect the side AB at P. If D is the midpoint of side AC, Then AB = 4AP. A Pp \D B 42. In the figure. Find the value of AB Where PT = Scm and PA = dem, TL oct P acm A 43. In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6cm, BC = 7em and CD = dem. Find AD. s Dy Cc A B P 44, Prove that “Ifa line touches a circle and from the point of contact a chord is drawn, the angle which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments.” 45. Prove that “If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtend by chord in the alternate segment, then the line is a tangent to the circle.” Scanned with CamScanner46. In figure. Land m are two parallel tangents at A and B, The tangent at C makes an intercept DE between the tangent I and m, Prove that ZDFE = 90" A Dy CY B —- ” 47. In figure, a circle is inscribed in a AABC having sides AB = 12 cm, BC = 8cm and AC = 10cm. Find AD, BE and CF. OR A circle is inscribed in a AABC having sides 8 cm, 10 cm and 12 cm as shown in fig. Find AD, BE and CF. . 48. If PA and PB are two tangents drawn from a point P to a circle with centre O touching it at A and B, prove that OP is the perpendicular bisector of AB. 49. If AABC is isosceles with AB = AC, Prove that the tangent at A to the circumeircle of AABC is parallel to BC. 50. Two circles intersect at A and B. From a point P on one of these circles, two lines segments PAC and PBD are drawn intersecting the other circles at C and D respectively. Prove that CD is parallel to the tangent at P. 51. Two circles intersect in points P and Q. A secant passing through P intersects the circles at A an B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, T and B are concyclic, 52. In the given figure TAS is a tangent to the circle, with centre O, at the point A. If ZOBA=32", find the value of x and y. Scanned with CamScanner3. In the given figure. PT is a tangent and PAB is a secant to a circle. If the bisector of ZATB intersect AB in M, Prove that: (i) ZPMT = ZPTM (ii) PT = PM T 54, In the adjoining figure, ABCD is a cyclic quadrilateral. AC is a diameter of the circle. MN is tangent to the circle at D, ZCAD = 40°, ZACB = 55° Determine ZADM and ZBAD 55. The diagonals of a parallelogram ABCD intersect at E. Show that the circumeircle of AADE and ABCE touch each other at E. 56. A circle is drawn with diameter AB interacting the hypotenuse AC of right triangle ABC at the point P. Show that the tangent to the circle at P bisects the side BC. 57. In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length 58. If AB, AC, PQ are tangents in below figure and AB = 5 cm, find the perimeter of AAPQ. 59. If PA and PB are tangents from an outside point P, such that PA = 10 em and ZAPB = 60" . Find the length of chord AB. 60. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of APCD. 61. Prove that the tangents at the extremities of any chord make equal angles with the chord. 62. From an external point P, two tangents PA and PB are drawn to the circle with centre O, Prove that OP is the perpendicular bisector of AB. 63. The radius of the incircle of a triangle is 4 cm and the segments into which one side divided by the point of contact are 6 cm and 8 cm, Find the other two sides of the triangle. 64. From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP = diameter of the circle, show that APB is an equilateral triangle. Scanned with CamScannered at B such that BC = 6 cm and AB = 8 cm. Find the 68. In fig. ABC is a right triangle right a radius of its incircle. 66. In the below figure, ABC is circumscribed, find the value of x. A 3cm | 3 i ) jure, quadrilateral ABCD is circumscribed, find the value of x. 67. In the above right-sided fi 68. In the below figure, quadrilateral ABCD is circumscribed, find the perimeter of quadrilateral ABCD. D Cy ea A\. IN DeE wr A, a} ——x eC 69, In the above right sided figure, quadrilateral ABCD is circumscribed and AD DC, find the value of x if the radius of incirele is 10 cm. ibed in a circle of radius 9 cm, 70. If an isosceles triangle ABC, in which AB = AC = 6 cm, is in find the area of the triangle. 71. A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the AABC. Scanned with CamScanner72. The tangent at a point C of a circle and a diameter AB when extended intersect at P. If ZPCA = 110°, find ZCBA Cc [> 73. In a right triangle ABC in which ZB = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC. 74, AB is a diameter and AC is a chord of a circle with centre O such that ZBAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC = BD. 75. In the below figure from an external point A, tangents AB and AC are drawn to a circle. PQ is a tangent to the circle at X. If AC = 15 cm, find the perimeter of the triangle APQ Scanned with CamScanner