Question

An equilateral triangle o side eam
eis inscribed in a circe. Find the

radius af the circle​

Answer 1

Question-

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

Solution-

For an equilateral triangle inscribed in a circle as shown above, the center of the circle is the circumcenter of the triangle.

For an equilateral triangle, because of symmetry,

∠ABO = ∠BOC = ∠AOC = 120°

$cos∠BOA = \frac{OB² + AO² - AB²}{2OB - AO} \\ \\ OB = AO = x \\ \\ cos(120) = \frac{ {2x}^{2} - {9}^{2} }{{2x}^{2}} \\ \\ cos(120) = −1/2 \\ \\ = \frac{ - 1}{2} = \frac{ {2x}^{2} - 81}{ {2x}^{2} } \\ \\ + {3x}^{2} = | - 8| \\ \\ {x}^{2} =81/3 \\ \\ {x}^{2} = 27 \\ \\ x = \sqrt{27} = 3 \sqrt{3}$