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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