If two tangents inclined at an angle of 60ᵒ are drawn to a circle of radius 3cm, then the length of each tangent is equal to

Other Exercises • • • • Mark the correct alternative in each of the following : Question 1. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 12 cm. Length PQ is (a) 12 cm (b) 13 cm (c) 8.5 cm (d) √119 cm Solution: (d) Radius of a circle OP = 5 cm OQ = 12 cm, PQ is tangent OP ⊥ PQ In right ∆OPQ, OQ² = OP² + PQ² (Pythagoras Theorem) => (12)² = (5)2 + PQ² => 144 = 25 + PQ² PQ² = 144 – 25 = 119 PQ = √119 Question 2. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (a) 7 cm (b) 12 cm (c) 15 cm (d) 24.5 cm Solution: (a) Let PQ be the tangent from Q to the circle with O as centre PQ = 24 cm OQ = 25 cm Let Radius OQ = r OQ ⊥ PQ Now in right ∆OPQ, OQ² = OP² + PQ² (Pythagoras Theorem) => (25)² = r² + (24)² => 625 = r² + 576 => r² = 625 – 576 = 49 = (7)² r = 7 Radius of the circle = 7 cm Question 3. The length of the tangent from a point A at a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is (a) √7 cm (b) 7 cm (c) 5 cm (d) 25 cm Solution: (c) Let AB be the tangent from A to the circle of centre O, then OB = 3 cm BA = 4 cm OB ⊥ BA In right ∆OBA, OA² = OB² + BA² (Pythagoras Theorem) = (3)² + (4)² = 9 + 16 = 25 = (5)² OA = 5 Distance of A from the centre O = 5 cm Question 4. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80° then ∠POA is equ...

(d) 3√3 cm Let AB and AC are tangent to circle with centre O and radius 3 Such that∠BAC = 60° ∵ OB = OC (radius of circle) AB = AC (lengths of tangents to circle from same external point) OA = OA (common side) ∴ΔOAB ~∠AOC (By SSS similarity criteria) ∴∠OAB =∠OAC (By property of similar triangles) ∴∠OAB = \(\frac\,tan \,30°\) ⇒ AB = OB cot 30° =√3 OB = 3√3 cm Hence, lengths of each tangent = 3√3 cm

Views: 5,953 152 x − 378 y answers. Had 5 marks been awarded for each correct answer and 2 marks been deducted for each wrong answer, then Raghav would have scored 80 marks. How many questions were there in the test? 29. Prove that: ( cosec θ − cot θ ) 2 = 1 + c o s θ 1 − c o s θ ​ 30. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. 3 OR In the given figure, find the perimeter of A BC, if A P = 12 cm. If two tangents inclined at an angle of 60ᵒ are drawn to a circle of radius 3cm, then the length of each tangent is equal to (a) 3√3 2 cm (b) 3cm (c) 6cm (d) 3√3cm Updated On Jan 1, 2023 Topic All topics Subject Mathematics Class Class 10 Answer Type Video solution: 1 Upvotes 128 Avg. Video Duration 4 min

Ans. 3. Two concentric circles are of radii 5cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. Ans. 4. Prove that the parallelogram circumscribing a circle is a rhombus. Ans. 5. In the figure given, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that –AOB = 90°. Ans. 6. In the given figure, OP is equal to diameter of the circle. Prove that ABP is an equilateral triangle. Ans. 7. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. Ans. 8. In the given figure, from an external point P, two tangents PQ and PR are drawn to a circle of radius 4 cm with centre O. If ∠QPR = 90°, then length of PQ is (a) 3 cm (b) 4 cm (c) 2 cm (d) 2√2 cm 9. Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. Ans. 10. From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB. Ans. 11. If the angle between two tangents drawn from an external point ‘P’ to a circle of radius ‘r’ and centre O is 60°, then find the length of OP. Ans. 12. If the radii of two concentric circles are 4 cm and 5 cm, then find the length of each chord...