2 Views

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: $1$


Team Careers360 2nd Jan, 2024
Answer (1)
Team Careers360 21st Jan, 2024

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 $\sqrt3$.

Know More About

Related Questions

TOEFL ® Registrations 2024
Apply
Accepted by more than 11,000 universities in over 150 countries worldwide
Manipal Online M.Com Admissions
Apply
Apply for Online M.Com from Manipal University
View All Application Forms

Download the Careers360 App on your Android phone

Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile

150M+ Students
30,000+ Colleges
500+ Exams
1500+ E-books