The chord of a unit (in cm ) circle subtends an angle of 120 degree at the centre . Find the length of the chord in cm.

Dear student,

Please find below the solution to the asked query:

Consider a circle with center O and with unit radius such that chord AB subtends an angle of 120° at the center.The figure is shown below:



Note that the perpendicular from the center to the chord bisects the chord.Let the length of chord AB is x cm, then we have   AD=DB=x2 cmAlso note that,  AOD=12AOB              =12×120°              =60°              =π3 radianIn right OAD, we have   ADOA=sinAOD   AD1=sinπ3   AD1=32    AD=32So the length of the chord is,  AB=2AD=2×32=3    

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  • 1
otherwise, if you don't give its radius
then
We know that chord and its angle at the centre is bisected by the altitude of
given triangle. so,angle at centre become 60°-60°.Now, in this way 2
triangles are formed . Now these are
congruent as you may know.
now,angle formed at the chord=30°-30°
(this angle is formed by radii at the centre )Now,considering one part of
congruent triangle having angles 90°,30° and 60°(90° is due to altitude )
This triangle is same as the triangle which is formed when equilateral triangle is bisected.so , side opp. to 60°=root3/2×a=root3/2×r.(which is half of chord)
so,length of the chord=root3/2×r×2
=root3×r
  • 1
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