Geometry
- Two chords AB and CD of a circle intersect at a point P inside the circle. If AP = 6 cm, PB = 4 cm and CP = 3 cm, find PD. Answer: D. 8 cm Answer: (D) 8 cm Intersecting chords theorem: AP × PB = CP × PD . 6 × 4 = 3 × PD ⇒ PD = 8 cm. 1
- In \(\triangle ABC\), \(\angle A = 80^\circ\). The bisectors of the exterior angles at B and C meet at O. Find \(\angle BOC\). Answer: B. 50^° Answer: (B) 50^° For the excentre formed by the external bisectors at B and C: ∠ BOC = 90^° - ∠ A/2 = 90^° - 40^° = 50^° . (... 1
- Each exterior angle of a regular polygon is 24°. How many sides does it have? Answer: D. 15 Answer: (D) 15 Number of sides = 360^°/24^° = 15 . 1
- A point is 13 cm from the centre of a circle of radius 5 cm. What is the length of the tangent drawn from the point to the circle? Answer: A. 12 cm Answer: (A) 12 cm The radius is perpendicular to the tangent: tangent = √(13² - 5²) = √(144) = 12 cm. 1
- In triangle ABC, I is the incentre and ∠A = 70°. Find ∠BIC. Answer: D. 125° Answer: (D) 125° ∠ BIC = 90^° + ∠ A/2 = 90^° + 35^° = 125^° . Reason: ∠ IBC + ∠ ICB = 1/2(180^° - 70^°) = 55^° , so ∠ BIC =... 1
- The angles of a triangle are in the ratio 2 : 3 : 4. What is the largest angle? Answer: A. 80° Answer: (A) 80° 2x + 3x + 4x = 180^° , so x = 20^° . Largest angle = 4 × 20 = 80^° . 1
- A chord of length 16 cm is at a distance of 6 cm from the centre of a circle. What is the radius of the circle? Answer: C. 10 cm Answer: (C) 10 cm The perpendicular from the centre bisects the chord, so half-chord = 8 cm. r = √(8² + 6²) = √(100) = 10 c... 1
- In triangle ABC, D and E are points on AB and AC such that DE ∥ BC. If AD = 4 cm and DB = 6 cm, what is the ratio of the area of triangle ADE to the area of trapezium DBCE? Answer: D. 4 : 21 Answer: (D) 4 : 21 Triangles ADE and ABC are similar with ratio AD : AB = 4 : 10 = 2 : 5, so their areas are in the ratio... 1