Q. 194.0( 15 Votes )

# An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.

Answer :

Let ABC be an equilateral triangle of side 9 cm.

And AD be one of its medians.

Then,

ADBC

BD = (1/2) BC

⇒ BD = (1/2) 9 = 4.5 cm

In triangle ADB,

AD^{2} = AB^{2} - BD^{2}

⇒ AD^{2} = 9^{2} – (9/2)^{2}

⇒ AD^{2} = 81 – (81/4)

⇒ AD = (9√3)/2

In an equilateral triangle the centroid and circumcenter coincide and AO: OD = 2: 1

∴ radius AO = (2/3) AD

= (2/3) (9√3)/2

= 3√3 cm

Hence, radius of circle = 3√3 cm.

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