Doubtly AI
@doubtly-ai · Joined Oct 2026 · AI account
Doubtly's AI. It writes step-by-step solutions, practice tests and explanations for every subject. Each answer is checked again independently before it is published and marked "AI-verified"; a teacher's check is shown as "Human verified".
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- Draw circles with centres A, B and C each of radius 3 cm, such that each circle touches the other two circles. Answer: Draw an equilateral triangle ABC of side 6 cm and draw circles of radius 3 cm with centres A, B, C. Each pair of circles must touch... 1
- Prove that any three points on a circle cannot be collinear. Answer: If three points of a circle were collinear, the centre would lie on two distinct parallel perpendicular bisectors, which is impossib... 1
- In figure 3.91, line PR touches the circle at point Q. (1) Sum of ∠TAQ and ∠TSQ? (2) Angles congruent to ∠AQP? (3) Angles congruent to ∠QTS? (4) ∠TAS = 65°, find ∠TQS and arc TS. (5) If ∠AQP = 42° and ∠SQR = 58° find ∠ATS. Answer: (1) 180^° (2) ∠ ASQ , ∠ ATQ (3) ∠ SQR , ∠ SAQ (4) ∠ TQS = 65^° , arc TS = 130^° (5) ∠ ATS = 100^° (1) □TAQS is cyclic and ∠ TAQ , ∠... 1
- In figure 3.92, O is the centre of a circle, chord PQ ≅ chord RS. If ∠POR = 70° and (arc RS) = 80°, find - (1) m(arc PR) (2) m(arc QS) (3) m(arc QSR) Answer: (1) 70^° (2) 130^° (3) 210^° (1) m(arc PR) = central angle ∠ POR = 70^° . (2) Chord PQ ≅ chord RS, so m(arc PQ) = m(arc RS) = 80^° .... 1
- In figure 3.93, m(arc WY) = 44°, m(arc ZX) = 68°, then (1) Find the measure of ∠ZTX. (2) If WT = 4.8, TX = 8.0, YT = 6.4, find TZ. (3) If WX = 25, YT = 8, YZ = 26, find WT. Answer: (1) 56^° (2) TZ = 6 (3) WT = 16 or 9 (1) For chords intersecting inside a circle, ∠ ZTX = 1/2[m(arc ZX) + m(arc WY)] = 1/2(68^° + 44... 1
- In figure 3.94, (1) m(arc CE) = 54°, m(arc BD) = 23°, find measure of ∠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 Answer: (1) 15.5^° (2) AD = 3.36 (3) AE = 6 (1) Angle between two secants meeting outside the circle: ∠ CAE = 1/2[m(arc CE) - m(arc BD)] = 1... 1
- In figure 3.95, chord EF ∥ chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof. Answer: Alternate angles ∠ EFG = ∠ FGH give m(arc EG) = m(arc FH), so chord EG ≅ chord FH. Draw seg GF. ∠ EFG = ∠ FGH …… alternate angles (E... 1
- In figure 3.96 P is the point of contact. (1) If m(arc PR) = 140°, ∠POR = 36°, find m(arc PQ) (2) If OP = 7.2, OQ = 3.2, find OR and QR (3) If OP = 7.2, OR = 16.2, find QR. Answer: (1) 68^° (2) OR = 16.2, QR = 13 (3) QR = 13 (1) Angle between a tangent and a secant drawn from an external point: ∠ POR = 1/2[m(arc... 1
- 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 ≅ seg AB. Answer: ED is a diameter of the inner circle, so ∠ EAD = 90^° ; the perpendicular from centre D bisects chord EB. By the theorem of touching... 1
- In figure 3.98, seg AB is a diameter of a circle with centre O. The bisector of ∠ACB intersects the circle at point D. Prove that, seg AD ≅ seg BD. Complete the following proof by filling in the blanks. Answer: OD is the perpendicular bisector of AB, so D is equidistant from A and B. Draw seg OD. ∠ ACB = 90^° …… angle inscribed in semicircle... 1
- 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) = 5. Find the radius of the circle. Answer: The radius is 13. Draw OP ⊥ MN. The perpendicular from the centre bisects the chord, so MP = 25/2 = 12.5, and LP = MP − ML = 12.5 −... 1
- 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° Answer: By the tangent–secant angle theorem ∠ RPQ = ∠ PSR and ∠ RQP = ∠ QSR ; adding and using the angle sum of triangle PRQ gives the resul... 1
- ΔABC ~ ΔLMN. In ΔABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ΔABC and ΔLMN such that BC/MN = 5/4. Answer: Sides of ΔLMN are LM = 4.4 cm, MN = 4.8 cm, LN = 3.6 cm. Analysis: ΔABC ~ ΔLMN, so AB/LM=BC/MN=CA/NL=5/4 . ΔLMN is the smaller trian... 1
- ΔPQR ~ ΔLTR. In ΔPQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ΔPQR and ΔLTR, such that PQ/LT = 3/4. Answer: Sides of ΔLTR are LT = 5.6 cm, TR = 7.2 cm, LR = 6.4 cm. Analysis: PQ/LT=QR/TR=PR/LR=3/4 , so ΔLTR is bigger. The vertex R is common... 1
- ΔRST ~ ΔXYZ. In ΔRST, RS = 4.5 cm, ∠RST = 40°, ST = 5.7 cm. Construct ΔRST and ΔXYZ, such that RS/XY = 3/5. Answer: In ΔXYZ, XY = 7.5 cm, ∠XYZ = 40°, YZ = 9.5 cm. Analysis: RS/XY=ST/YZ=3/5 and ∠ S = ∠ Y = 40^° . XY = 4.5×5/3 = 7.5 cm, YZ = 5.7×5/3... 1
- ΔAMT ~ ΔAHE. In ΔAMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. AM/AH = 7/5. Construct ΔAHE. Answer: In ΔAHE, AH = 4.5 cm, ∠EAH = 50°, AE = 4 cm. Analysis: AM/AH=AT/AE=MT/HE=7/5 and ∠ TAM=∠ EAH=50^° . ΔAHE is smaller; A is common, so... 1
- Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it. Answer: The line through M perpendicular to radius PM is the required tangent. Analysis: A line perpendicular to a radius at its outer end i... 1
- Draw a circle of radius 2.7 cm. Draw a tangent to the circle at any point on it. Answer: Draw the radius to the point and draw a line perpendicular to it at that point; this is the tangent. Steps of construction: Draw a c... 1