Geometry · Textbook solutions

Circle

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 70 questions

Tangent segment theorem

2 q

Solved Examples

Worked · 2
  1. Tangent segment theorem SolvedEx.1
    In the adjoining figure circle with centre D touches the sides of ∠ACB\angle ACB at A and B. If ∠ACB=52∘\angle ACB = 52^\circ, find measure of ∠ADB\angle ADB.
  2. Tangent segment theorem SolvedEx.2
    Point O is the centre of a circle. Line aa and line bb are parallel tangents to the circle at P and Q. Prove that segment PQ is a diameter of the circle.

Practice set 3.1

4 q
  1. Ex 3.1 Q.1
    In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a tangent at A. Answer the following questions. (1) What is the measure of ∠CAB\angle CAB ? Why ? (2) What is the distance of point C from line AB? Why ? (3) d(A,B)=6d(\text{A},\text{B}) = 6 cm, find d(B,C)d(\text{B},\text{C}). (4) What is the measure of ∠ABC\angle ABC ? Why ?
  2. Ex 3.1 Q.2
    In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then (1) What is the length of each tangent segment ? (2) What is the measure of ∠MRO\angle MRO ? (3) What is the measure of ∠MRN\angle MRN ?
  3. Ex 3.1 Q.3
    Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN\angle MRN as well as ∠MON\angle MON.
  4. Ex 3.1 Q.4
    What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.

Practice set 3.2

5 q
  1. Ex 3.2 Q.1
    Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.
  2. Ex 3.2 Q.2
    Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres.
  3. Ex 3.2 Q.3
    If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other - (i) externally (ii) internally.
  4. Ex 3.2 Q.4
    In fig 3.27, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that - (1) seg AP ∥\parallel seg BQ, (2) △APR∼△RQB\triangle APR \sim \triangle RQB, and (3) Find ∠RQB\angle RQB if ∠PAR=35∘\angle PAR = 35^\circ
  5. Ex 3.2 Q.5
    In fig 3.28 the circles with centres A and B touch each other at E. Line ll is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm.

Property of sum of measures of arcs

2 q

Solved Examples

Worked · 2
  1. Property of sum of measures of arcs SolvedEx.1
    A, B, C are any points on the circle with centre O. (i) Write the names of all arcs formed due to these points. (ii) If mm arc (BC) =110∘= 110^\circ and mm arc (AB) =125∘= 125^\circ, find measures of all remaining arcs.
  2. Property of sum of measures of arcs SolvedEx.2
    In the figure 3.36 a rectangle PQRS is inscribed in a circle with centre T. Prove that, (i) arc PQ ≅\cong arc SR (ii) arc SPQ ≅\cong arc PQR

Practice set 3.3

3 q
  1. Ex 3.3 Q.1
    In figure 3.37, points G, D, E, F are concyclic points of a circle with centre C. ∠ECF=70∘\angle ECF = 70^\circ, m(arc DGF)=200∘m(\text{arc DGF}) = 200^\circ find m(arc DE)m(\text{arc DE}) and m(arc DEF)m(\text{arc DEF}).
  2. Ex 3.3 Q.2
    In fig 3.38 △QRS\triangle QRS is an equilateral triangle. Prove that, (1) arc RS ≅\cong arc QS ≅\cong arc QR (2) m(arc QRS)=240∘m(\text{arc QRS}) = 240^\circ.
  3. Ex 3.3 Q.3
    In fig 3.39 chord AB ≅\cong chord CD, Prove that, arc AC ≅\cong arc BD

Converse of cyclic quadrilateral theorem

3 q

Solved Examples

Worked · 3
  1. Converse of cyclic quadrilateral theorem SolvedEx.1
    In figure 3.51, chord LM ≅\cong chord LN, ∠L=35∘\angle L = 35^\circ find (i) m(arc MN)m(\text{arc MN}) (ii) m(arc LN)m(\text{arc LN})
  2. Converse of cyclic quadrilateral theorem SolvedEx.2
    In figure 3.52, chords PQ and RS intersect at T. (i) Find m(arc SQ)m(\text{arc SQ}) if m∠STQ=58∘m\angle STQ = 58^\circ, m∠PSR=24∘m\angle PSR = 24^\circ. (ii) Verify, ∠STQ=12[m(arc PR)+m(arc SQ)]\angle STQ = \dfrac{1}{2}\left[m(\text{arc PR}) + m(\text{arc SQ})\right] (iii) Prove that : ∠STQ=12[m(arc PR)+m(arc SQ)]\angle STQ = \dfrac{1}{2}\left[m(\text{arc PR}) + m(\text{arc SQ})\right] for any measure of ∠STQ\angle STQ. (iv) Write in words the property in (iii).
  3. Converse of cyclic quadrilateral theorem SolvedEx.3
    Prove that, if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.

Practice set 3.4

8 q
  1. Ex 3.4 Q.1
    In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following. (1) ∠AOB\angle AOB (2) ∠ACB\angle ACB (3) arc AB (4) arc ACB.
  2. Ex 3.4 Q.2
    In figure 3.57, □\squarePQRS is cyclic. side PQ ≅\cong side RQ. ∠PSR=110∘\angle PSR = 110^\circ, Find- (1) measure of ∠PQR\angle PQR (2) m(arc PQR)m(\text{arc PQR}) (3) m(arc QR)m(\text{arc QR}) (4) measure of ∠PRQ\angle PRQ
  3. Ex 3.4 Q.3
    □\squareMRPN is cyclic, ∠R=(5x−13)∘\angle R = (5x - 13)^\circ, ∠N=(4x+4)∘\angle N = (4x + 4)^\circ. Find measures of ∠R\angle R and ∠N\angle N.
  4. Ex 3.4 Q.4
    In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠RTS\angle RTS is an acute angle.
  5. Ex 3.4 Q.5
    Prove that, any rectangle is a cyclic quadrilateral.
  6. Ex 3.4 Q.6
    In figure 3.59, altitudes YZ and XT of △WXY\triangle WXY intersect at P. Prove that, (1) □\squareWZPT is cyclic. (2) Points X, Z, T, Y are concyclic.
  7. Ex 3.4 Q.7
    In figure 3.60, m(arc NS)=125∘m(\text{arc NS}) = 125^\circ, m(arc EF)=37∘m(\text{arc EF}) = 37^\circ, find the measure ∠NMS\angle NMS.
  8. Ex 3.4 Q.8
    In figure 3.61, chords AC and DE intersect at B. If ∠ABE=108∘\angle ABE = 108^\circ, m(arc AE)=95∘m(\text{arc AE}) = 95^\circ, find m(arc DC)m(\text{arc DC}).

Tangent secant segments theorem

4 q

Solved Examples

Worked · 4
  1. Tangent secant segments theorem SolvedEx.1
    In figure 3.73, seg PS is a tangent segment. Line PR is a secant. If PQ = 3.6, QR = 6.4, find PS.
  2. Tangent secant segments theorem SolvedEx.2
    In figure 3.74, chord MN and chord RS intersect each other at point P. If PR = 6, PS = 4, MN = 11 find PN.
  3. Tangent secant segments theorem SolvedEx.3
    In figure 3.75, two circles intersect each other in points X and Y. Tangents drawn from a point M on line XY touch the circles at P and Q. Prove that, seg PM ≅\cong seg QM.
  4. Tangent secant segments theorem SolvedEx.4
    In figure 3.76, seg PQ is a diameter of a circle with centre O. R is any point on the circle. seg RS ⊥\perp seg PQ. Prove that, SR is the geometric mean of PS and SQ. [That is, SR2=PS×SQSR^2 = PS \times SQ]

Practice set 3.5

5 q
  1. Ex 3.5 Q.1
    In figure 3.77, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS.
  2. Ex 3.5 Q.2
    In figure 3.78, chord MN and chord RS intersect at point D. (1) If RD = 15, DS = 4, MD = 8 find DN (2) If RS = 18, MD = 9, DN = 8 find DS
  3. Ex 3.5 Q.3
    In figure 3.79, O is the centre of the circle and B is a point of contact. seg OE ⊥\perp seg AD, AB = 12, AC = 8, find (1) AD (2) DC (3) DE.
  4. Ex 3.5 Q.4
    In figure 3.80, if PQ = 6, QR = 10, PS = 8 find TS.
  5. Ex 3.5 Q.5
    In figure 3.81, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is rr. Prove that, DE×GE=4r2DE \times GE = 4r^2

Problem set 3

34 q
  1. Four alternative answers for each of the following questions are given. Choose the correct alternative.
    PS3 Q.1 (1)
    Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers ?
    1. A.
      4.4 cm
    2. B.
      8.8 cm
    3. C.
      2.2 cm
    4. D.
      8.8 or 2.2 cm
  2. PS3 Q.1 (2)
    Two circles intersect each other such that each circle passes through the centre of the other. If the distance between their centres is 12, what is the radius of each circle ?
    1. A.
      6 cm
    2. B.
      12 cm
    3. C.
      24 cm
    4. D.
      can't say
  3. PS3 Q.1 (3)
    A circle touches all sides of a parallelogram. So the parallelogram must be a, ................. .
    1. A.
      rectangle
    2. B.
      rhombus
    3. C.
      square
    4. D.
      trapezium
  4. PS3 Q.1 (4)
    Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.
    1. A.
      25 cm
    2. B.
      24 cm
    3. C.
      7 cm
    4. D.
      14 cm
  5. PS3 Q.1 (5)
    If two circles are touching externally, how many common tangents of them can be drawn?
    1. A.
      One
    2. B.
      Two
    3. C.
      Three
    4. D.
      Four
  6. PS3 Q.1 (6)
    ∠ACB\angle ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB=65∘\angle ACB = 65^\circ, find m(arc ACB)m(\text{arc ACB}).
    1. A.
      65∘65^\circ
    2. B.
      130∘130^\circ
    3. C.
      295∘295^\circ
    4. D.
      230∘230^\circ
  7. PS3 Q.1 (7)
    Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED.
    1. A.
      7
    2. B.
      8
    3. C.
      11.2
    4. D.
      9
  8. PS3 Q.1 (8)
    In a cyclic □\squareABCD, twice the measure of ∠A\angle A is thrice the measure of ∠C\angle C. Find the measure of ∠C\angle C?
    1. A.
      36
    2. B.
      72
    3. C.
      90
    4. D.
      108
  9. PS3 Q.1 (9)
    Points A, B, C are on a circle, such that m(arc AB)=m(arc BC)=120∘m(\text{arc AB}) = m(\text{arc BC}) = 120^\circ. No point, except point B, is common to the arcs. Which is the type of △ABC\triangle ABC?
    1. A.
      Equilateral triangle
    2. B.
      Scalene triangle
    3. C.
      Right angled triangle
    4. D.
      Isosceles triangle
  10. PS3 Q.1 (10)
    Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ\angle XYZ is an acute angle. (ii) ∠XYZ\angle XYZ can't be a right angle. (iii) ∠XYZ\angle XYZ is an obtuse angle. (iv) Can't make a definite statement for measure of ∠XYZ\angle XYZ.
    1. A.
      Only one
    2. B.
      Only two
    3. C.
      Only three
    4. D.
      All
  11. PS3 Q.2
    Line ll touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following. (1) What is d(O,P)=?d(\text{O}, \text{P}) = ? Why ? (2) If d(O,Q)=8d(\text{O}, \text{Q}) = 8 cm, where does the point Q lie ? (3) If d(PQ)=15d(\text{PQ}) = 15 cm, How many locations of point R are line on line ll ? At what distance will each of them be from point P ?
  12. PS3 Q.3
    In figure 3.83, M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = 636\sqrt{3} then find - (1) Radius of the circle. (2) Measures of ∠K\angle K and ∠M\angle M.
  13. PS3 Q.4
    In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle is rr and l(AB)=rl(\text{AB}) = r, Prove that, □\squareABOC is a square.
  14. PS3 Q.5
    In figure 3.85, □\squareABCD is a parallelogram. It circumscribes the circle with cnetre T. Point E, F, G, H are touching points. If AE = 4.5, EB = 5.5, find AD.
  15. PS3 Q.6
    In figure 3.86, circle with centre M touches the circle with centre N at point T. Radius RM touches the smaller circle at S. Radii of circles are 9 cm and 2.5 cm. Find the answers to the following questions hence find the ratio MS:SR. (1) Find the length of segment MT (2) Find the length of seg MN (3) Find the measure of ∠NSM\angle NSM.
  16. PS3 Q.7
    In the adjoining figure circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that, radius XA ∥\parallel radius YB. Fill in the blanks and complete the proof.
  17. PS3 Q.8
    In figure 3.88, circles with centres X and Y touch internally at point Z. Seg BZ is a chord of bigger circle and it itersects smaller circle at point A. Prove that, seg AX ∥\parallel seg BY.
  18. PS3 Q.9
    In figure 3.89, line ll touches the circle with centre O at point P. Q is the mid point of radius OP. RS is a chord through Q such that chords RS ∥\parallel line ll. If RS = 12 find the radius of the circle.
  19. PS3 Q.10
    In figure 3.90, seg AB is a diameter of a circle with centre C. Line PQ is a tangent, which touches the circle at point T. seg AP ⊥\perp line PQ and seg BQ ⊥\perp line PQ. Prove that, seg CP ≅\cong seg CQ.
  20. PS3 Q.11
    Draw circles with centres A, B and C each of radius 3 cm, such that each circle touches the other two circles.
  21. PS3 Q.12
    Prove that any three points on a circle cannot be collinear.
  22. PS3 Q.13
    In figure 3.91, line PR touches the circle at point Q. Answer the following questions with the help of the figure. (1) What is the sum of ∠TAQ\angle TAQ and ∠TSQ\angle TSQ ? (2) Find the angles which are congruent to ∠AQP\angle AQP. (3) Which angles are congruent to ∠QTS\angle QTS ? (4) ∠TAS=65∘\angle TAS = 65^\circ, find the measure of ∠TQS\angle TQS and arc TS. (5) If ∠AQP=42∘\angle AQP = 42^\circ and ∠SQR=58∘\angle SQR = 58^\circ find measure of ∠ATS\angle ATS.
  23. PS3 Q.14
    In figure 3.92, O is the centre of a circle, chord PQ ≅\cong chord RS. If ∠POR=70∘\angle POR = 70^\circ and (arc RS) =80∘= 80^\circ, find - (1) m(arc PR)m(\text{arc PR}) (2) m(arc QS)m(\text{arc QS}) (3) m(arc QSR)m(\text{arc QSR})
  24. PS3 Q.15
    In figure 3.93, m(arc WY)=44∘m(\text{arc WY}) = 44^\circ, m(arc ZX)=68∘m(\text{arc ZX}) = 68^\circ, then (1) Find the measure of ∠ZTX\angle ZTX. (2) If WT = 4.8, TX = 8.0, YT = 6.4, find TZ. (3) If WX = 25, YT = 8, YZ = 26, find WT.
  25. PS3 Q.16
    In figure 3.94, (1) m(arc CE)=54∘m(\text{arc CE}) = 54^\circ, m(arc BD)=23∘m(\text{arc BD}) = 23^\circ, find measure of ∠CAE\angle CAE. (2) If AB = 4.2, BC = 5.4, AE = 12.0, find AD (3) If AB = 3.6, AC = 9.0, AD = 5.4, find AE
  26. PS3 Q.17
    In figure 3.95, chord EF ∥\parallel chord GH. Prove that, chord EG ≅\cong chord FH. Fill in the blanks and write the proof.
  27. PS3 Q.18
    In figure 3.96 P is the point of contact. (1) If m(arc PR)=140∘m(\text{arc PR}) = 140^\circ, ∠POR=36∘\angle POR = 36^\circ, find m(arc PQ)m(\text{arc PQ}) (2) If OP = 7.2, OQ = 3.2, find OR and QR (3) If OP = 7.2, OR = 16.2, find QR.
  28. PS3 Q.19
    In figure 3.97, circles with centres C and D touch internally at point E. D lies on the inner circle. Chord EB of the outer circle intersects inner circle at point A. Prove that, seg EA ≅\cong seg AB.
  29. PS3 Q.20
    In figure 3.98, seg AB is a diameter of a circle with centre O. The bisector of ∠ACB\angle ACB intersects the circle at point D. Prove that, seg AD ≅\cong seg BD. Complete the following proof by filling in the blanks.
  30. PS3 Q.21
    In figure 3.99, seg MN is a chord of a circle with centre O. MN = 25, L is a point on chord MN such that ML = 9 and d(O,L)=5d(\text{O},\text{L}) = 5. Find the radius of the circle.
  31. PS3 Q.22
    In figure 3.100, two circles intersect each other at points S and R. Their common tangent PQ touches the circle at points P, Q. Prove that, ∠PRQ+∠PSQ=180∘\angle PRQ + \angle PSQ = 180^\circ
  32. PS3 Q.23
    In figure 3.101, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R, S and P, Q respectively. Prove that : seg SQ ∥\parallel seg RP.
  33. PS3 Q.24
    In figure 3.102, two circles intersect each other at points A and E. Their common secant through E intersects the circles at points B and D. The tangents of the circles at points B and D intersect each other at point C. Prove that □\squareABCD is cyclic.
  34. PS3 Q.25
    In figure 3.103, seg AD ⊥\perp side BC, seg BE ⊥\perp side AC, seg CF ⊥\perp side AB. Ponit O is the orthocentre. Prove that, point O is the incentre of △DEF\triangle DEF.