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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- 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
- Ratio of areas of two triangles with equal heights is 2 : 3. If base of the smaller triangle is 6 cm then what is the corresponding base of the bigger triangle? Answer: The base of the bigger triangle is 9 cm. Areas of triangles with equal heights are proportional to their bases. Let the base of the... 1
- In figure 1.73, ∠ABC = ∠DCB = 90°, AB = 6, DC = 8 then A(ΔABC)/A(ΔDCB) = ? Answer: A(triangle ABC)/A(triangle DCB) = 3/4 Both triangles have the same base BC. Their heights on this base are AB = 6 (since AB ⊥ BC) an... 1
- In figure 1.74, PM = 10 cm, A(ΔPQS) = 100 sq.cm, A(ΔQRS) = 110 sq.cm then find NR. Answer: NR = 11 cm triangle PQS and triangle QRS have the common base QS. Their heights are PM and NR respectively. Areas of triangles with... 1
- ΔMNT ~ ΔQRS. Length of altitude drawn from point T is 5 and length of altitude drawn from point S is 9. Find the ratio A(ΔMNT)/A(ΔQRS). Answer: A(triangle MNT)/A(triangle QRS) = 25/81 Since triangle MNT sim triangle QRS , vertex T corresponds to vertex S, so the altitudes fro... 1
- In figure 1.75, A-D-C and B-E-C, seg DE ∥ side AB. If AD = 5, DC = 3, BC = 6.4 then find BE. Answer: BE = 4 Let BE = x, then EC = 6.4 − x. In triangle ABC , DE ∥ AB, so by the basic proportionality theorem, CD/DA = CE/EB 3/5 = 6.4 -... 1
- In the figure 1.76, seg PA, seg QB, seg RC and seg SD are perpendicular to line AD. AB = 60, BC = 70, CD = 80, PS = 280 then find PQ, QR and RS. Answer: PQ = 80, QR = 280/3 , RS = 320/3 Lines perpendicular to the same line are parallel, so PA ∥ QB ∥ RC ∥ SD. Parallel lines cut two tra... 1
- In ΔPQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY ∥ QR. Answer: By the angle bisector theorem PX/XQ = MP/MQ and PY/YR = MP/MR ; since MQ = MR these are equal, so XY ∥ QR. In triangle PMQ , ray MX... 1
- In fig 1.78, bisectors of ∠B and ∠C of ΔABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find AX/XY. Answer: AX/XY = 3/2 In triangle ABY , BX bisects ∠ B , so by the angle bisector theorem AX/XY = AB/BY In triangle ACY , CX bisects ∠ C , so... 1
- In □ABCD, seg AD ∥ seg BC. Diagonal AC and diagonal BD intersect each other in point P. Then show that AP/PD = PC/BP. Answer: triangle APD sim triangle CPB (AA test), which gives AP/PD = PC/BP . In triangle APD and triangle CPB : ∠ PAD cong ∠ PCB (alternate... 1
- In fig 1.80, XY ∥ seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC. Answer: AC = 22.5 2AX = 3BX, so AX/BX = 3/2 . By componendo, AX + BX/BX = 3 + 2/2 , so AB/BX = 5/2 …… (I) In triangle BCA and triangle BYX ,... 1