Question : Chord PQ is the perpendicular bisector of radius OA of the circle with centre O. (A is a point on the edge of the circle). If the length of Arc $PAQ=\frac{2\pi}{3}$. What is the length of chord PQ?
Option 1: $2$
Option 2: $\sqrt{3}$
Option 3: $2\sqrt{3}$
Option 4: $1$
Correct Answer: $\sqrt{3}$
Solution : PQ is perpendicular bisector of OA. $\therefore$ OP = OQ = PA = AQ $\therefore$ OPAQ is a rhombus. As we know the angle subtended at the centre by an arc is twice that at the circumference. Thus, 2 $\angle$PAQ = Reflex $\angle$POQ ⇒ 2 $\angle$PAQ = 360° – $\angle$POQ ⇒ 3$\angle$ PAQ = 360° ($\because\angle$PAQ = $\angle$POQ) $\therefore \angle$ PAQ = 120$^\circ$ = $\angle$ POQ = $\frac{2\pi}{3}$ Again, Radius (r) = $\frac{\text{arc length}}{\theta}$ = $\frac{\frac{2\pi}{3}}{\frac{2\pi}{3}}$ = 1 Now, in $\triangle$ OPB, OP = 1 unit $\angle$POB = 60° $\therefore$ sin 60° = $\frac{PB}{OP}$ ⇒ PB = $\frac{\sqrt3}{2}$ $\therefore$ PQ = 2 × $\frac{\sqrt3}{2}=\sqrt3$ Hence, the correct answer is $\sqrt3$.
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Question : Out of two concentric circles, the radius of the outer circle is 6 cm and the chord PQ of the length 10 cm is a tangent to the inner circle. Find the radius (in cm) of the inner circle.
Option 1: $4$
Option 2: $\sqrt{7}$
Option 3: $\sqrt{13}$
Option 4: $\sqrt{11}$
Question : Let C be a circle with centre O and radius 5 cm. Let PQ be a tangent to the circle and A be the point of tangency. Let B be a point on PQ such that the length of AB is 12 cm. If the line joining O and B intersects the circle at R, find the length of BR (in cm).
Option 1: 2
Option 2: 13
Option 3: 6
Option 4: 8
Question : AB is a chord in a circle of radius 13 cm. From centre O, a perpendicular is drawn through AB intersecting AB at point C. The length of OC is 5 cm. What is the length of AB?
Option 1: 24 cm
Option 2: 12 cm
Option 3: 20 cm
Option 4: 15 cm
Question : The area of a circle is the same as the area of a square. What is the ratio of the diameter of the circle and the diagonal of the square?
Option 1: $1:\sqrt{\pi }$
Option 2: $2:\sqrt{\pi }$
Option 3: $\sqrt{2}:\sqrt{\pi }$
Option 4: $1:{\pi }$
Question : AB is a chord of a circle having a radius of 1.7 cm. If the distance of this chord AB from the centre of the circle is 0.8 cm, then what is the length (in cm) of the chord AB?
Option 1: 4
Option 2: 1
Option 3: 3
Option 4: 2
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