Angles Subtended by Arcs • Class 9 Mathematics • Ganita Manjari • Chapter 5
Practice Mode: पहले question खुद solve करें। समझ न आए तो हिंदी में समझें और फिर Solution देखें से पूरा solution check करें।
Question 1
In a circle with centre O, the central angle ∠AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?
हिंदी में समझें: OA और OB radii हैं, इसलिए OA = OB = 12 cm। Triangle AOB isosceles है। Centre angle 60° होने पर बाकी दोनों angles भी 60° होंगे, इसलिए triangle equilateral होगा और AB = 12 cm.
Step-by-Step Solution:
Central angle and chord AB
OA = OB = 12 cm.
∠OAB = ∠OBA = x 60° + x + x = 180° 2x = 120° x = 60°
All three angles are 60°, so ΔAOB is equilateral.
Final Answer: AB = 12 cm.
Question 2
Let A and B be two points on a circle with centre O.
(i) Are there points X, Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB?
(ii) Is it true that if ∠AXB = ∠AYB, then X and Y lie on the same side of the circle?
(iii) If ∠AXB = ∠AYB, and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y?
हिंदी में समझें: Same chord AB के same segment में बने angles बराबर होते हैं। लेकिन केवल angle equality से same side साबित नहीं होती। Same-side condition के साथ equal angles होने पर चारों points concyclic होते हैं।
Step-by-Step Solution:
Same chord AB subtending equal angles
(i) No. Points X and Y on the same side of AB lie in the same segment, so angles subtended by chord AB are equal.
(ii) No. Equality alone does not always imply same side. For example, if AB is a diameter, points on opposite sides can both subtend 90°.
(iii) Yes, provided X and Y are on the same side of AB. By the converse of the same-segment result, equal angles imply A, B, X and Y are concyclic.
Final Answers: (i) No. (ii) No, not necessarily. (iii) Yes, with the same-side condition.
Question 3
Find x in Fig. 5.26.
हिंदी में समझें: Fig. 5.26 में ABCD cyclic quadrilateral है। ∠ADC = 100° और opposite angle ∠ABC = x है। Cyclic quadrilateral में opposite angles का sum 180° होता है।