All questions
- Prove that, any rectangle is a cyclic quadrilateral. Answer: Opposite angles of a rectangle are 90^° + 90^° = 180^° , so by the converse of the cyclic quadrilateral theorem it is cyclic. Let □A... 1
- In figure 3.59, altitudes YZ and XT of ΔWXY intersect at P. Prove that, (1) □WZPT is cyclic. (2) Points X, Z, T, Y are concyclic. Answer: (1) ∠ WZP + ∠ WTP = 180^° ; (2) seg XY subtends equal right angles at Z and T on the same side of it. (1) YZ ⊥ WX, so ∠ WZP = 90^° .... 1
- In figure 3.60, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠NMS. Answer: ∠ NMS = 44^° When two secants meet outside a circle, the angle between them is half the difference of the intercepted arcs: ∠ NMS =... 1
- In figure 3.61, chords AC and DE intersect at B. If ∠ABE = 108°, m(arc AE) = 95°, find m(arc DC). Answer: m(arc DC) = 121^° When two chords intersect inside a circle, each angle formed is half the sum of the arcs intercepted by the angle... 1
- In figure 3.77, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS. Answer: PS = 18 and RS = 10 By the tangent secant segments theorem, PQ² = PR × PS 144 = 8 × PS PS = 18 RS = PS - PR = 18 - 8 = 10 1
- 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 Answer: (1) DN = 7.5 (2) DS = 12 or 6 By the theorem of chords intersecting inside a circle, RD × DS = MD × DN . (1) 15 × 4 = 8 × DN DN = 60... 1
- In figure 3.79, O is the centre of the circle and B is a point of contact. seg OE ⊥ seg AD, AB = 12, AC = 8, find (1) AD (2) DC (3) DE. Answer: (1) AD = 18 (2) DC = 10 (3) DE = 5 (1) By the tangent secant segments theorem, AB² = AC × AD 144 = 8 × AD, AD = 18 (2) DC = AD − AC... 1
- In figure 3.80, if PQ = 6, QR = 10, PS = 8 find TS. Answer: TS = 4 PR = PQ + QR = 6 + 10 = 16. By the theorem of secants intersecting outside a circle, PS × PT = PQ × PR 8 × PT = 6 × 16 = 96 P... 1
- In figure 3.81, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, DE × GE = 4r² Answer: triangle EFD is right angled at F and FG ⊥ ED, so EF² = EG × ED , i.e. (2r)² = DE × GE . Draw seg FG. ∠ EFD = 90^° (tangent is perpe... 1
- Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers? (A) 4.4 cm (B) 8.8 cm (C) 2.2 cm (D) 8.8 or 2.2 cm Answer: (D) 8.8 or 2.2 cm The question does not say how the circles touch. Touching externally: distance = 5.5 + 3.3 = 8.8 cm. Touching inte... 1
- 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? (A) 6 cm (B) 12 cm (C) 24 cm (D) can't say Answer: (B) 12 cm Let the centres be A and B. The circle with centre A passes through B, so its radius = AB = 12. Similarly the circle with... 1
- A circle touches all sides of a parallelogram. So the parallelogram must be a, ........ (A) rectangle (B) rhombus (C) square (D) trapezium Answer: (B) rhombus Let □ABCD touch the circle. Tangent segments from an external point are equal, which gives AB + CD = AD + BC. In a paral... 1
- 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. (A) 25 cm (B) 24 cm (C) 7 cm (D) 14 cm Answer: (C) 7 cm The radius is perpendicular to the tangent at the point of contact, so the radius r, the tangent segment and the distance f... 1
- If two circles are touching externally, how many common tangents of them can be drawn? (A) One (B) Two (C) Three (D) Four Answer: (C) Three Two circles touching externally have two direct common tangents, one on each side of the line of centres, and one more com... 1
- ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB). (A) 65° (B) 130° (C) 295° (D) 230° Answer: (D) 230^° An inscribed angle is half the measure of its intercepted arc. ∠ ACB intercepts arc AB (the arc not containing C): m(arc A... 1
- Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED. (A) 7 (B) 8 (C) 11.2 (D) 9 Answer: (A) 7 By the theorem of chords intersecting inside a circle, AE × EB = CE × ED 5.6 × 10 = 8 × ED ED = 56/8 = 7 1
- In a cyclic □ABCD, twice the measure of ∠A is thrice the measure of ∠C. Find the measure of ∠C? (A) 36 (B) 72 (C) 90 (D) 108 Answer: (B) 72 2∠A = 3∠C, so ∠ A = 3/2∠ C . Opposite angles of a cyclic quadrilateral are supplementary: 3/2∠ C + ∠ C = 180^° 5/2∠ C = 180^°... 1
- Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ΔABC? (A) Equilateral (B) Scalene (C) Right angled (D) Isosceles Answer: (A) Equilateral triangle The arcs AB and BC have only B in common, so the remaining arc CA has measure 360^° - 120^° - 120^° = 120^°... 1
- Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements about ∠XYZ are true? (A) Only one (B) Only two (C) Only three (D) All Answer: (C) Only three For a point on the circle, the angle subtended by the diameter XZ is 90^° . For a point Y in the interior of the circ... 1
- Line l touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following: d(O, P), position of Q if d(O, Q) = 8 cm, and locations of point R on line l. Answer: (1) 9 cm (2) in the interior of the circle (3) two locations, each 12 cm from P (1) d(O, P) = 9 cm, because OP is a radius (P is the... 1
- In figure 3.83, M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = 6√3 then find - (1) Radius of the circle. (2) Measures of ∠K and ∠M. Answer: (1) Radius = 6 (2) ∠ K = 30^° , ∠ M = 60^° (1) ∠ MLK = 90^° (tangent ⊥ radius). In right-angled triangle MLK : ML² = MK² - KL² = 144... 1
- In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle is r and l(AB) = r, Prove that, □ABOC is a square. Answer: All four sides of □ABOC equal r and ∠ ABO = 90^° , so it is a square. AB = r (given) and AC = AB = r (tangent segments from an exter... 1
- In figure 3.85, □ABCD is a parallelogram. It circumscribes the circle with centre T. Point E, F, G, H are touching points. If AE = 4.5, EB = 5.5, find AD. Answer: AD = 10 Tangent segments from an external point are congruent: AE = AH, BE = BF, CF = CG, DG = DH. Adding, AE + BE + CG + DG = AH +... 1
- 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 (1) MT (2) MN (3) ∠NSM, hence find the ratio MS:SR. Answer: (1) 9 cm (2) 6.5 cm (3) 90^° ; MS : SR = 2 : 1 (1) T is on the bigger circle, so MT is its radius: MT = 9 cm. (2) The circles touch... 1