Question

IN FIGURE O IS THE CENTRE OF THE CIRCLE angle AOB= angle BOC = angle COA and AO=2root3cm.Find the perimeter of triangle ABC

Answer 1

∠AOB = ∠BOC = ∠COA = 120°  (as their sum is 360° )

Angle made by chord AB at O = Angle made by chord BC at O
                  = angle made by chord CA  at O

Hence, AB = BC = CA.                 So ΔABC is equilateral.
O is the circumcenter of equilateral ΔABC.   
AO = radius = 2√3 cm = 2/3 * altitude from A
        (O is also the centroid)
Altitude from A =   √3/2 * AB
So   2/3 * √3/2 * AB = 2√3 cm
                            AB = 6 cm

Hence the perimeter = 3 * 6 = 18 cm

Answer 2

Step-by-step explanation:

∠AOB = ∠BOC = ∠COA = 120°

So, AB = BC = CA and ΔABC is equilateral.

O is the circumcenter of equilateral ΔABC.    

AO = radius = 2√3 cm = $\frac{2}{3}$ of altitude from A  

Altitude from A =   √3/2 * AB

So 2/3 * √3/2 * AB = 2√3 cm

                         AB = 6 cm

∴ Perimeter = 3 * 6 = 18 cm

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