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

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An equilateral triangle of side $9 c m$ is inscribed in a circle. Find the radius of the circle.

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An equilateral triangle of side

9 c m

is inscribed in a circle. Find the radius of the circle.

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Text solutionVerified

Consider

△ A BC

as an equilateral triangle with side

9 c m

Take

A D

as one of its median

We know that

A D ⊥ BC

It can be written as

B D = \frac{1}{2} \times BC

By substituting the values

B D = \frac{1}{2} \times 9

So we get

B D = 4.5 c m

Consider

△ A D B

Using the Pythagoras theorem

A B^{2} = A D^{2} + B D^{2}

Substituting the values

9^{2} = A D^{2} + (9/2)^{2}

On further calculation

A D^{2} = 9^{2} - (\frac{9}{2})^{2}

So we get

A D^{2} = 81 - \frac{81}{4}

By taking out the square root

A D = \frac{9 3}{2} c m

We know that the centroid and circumcenter coincide in an equilateral triangle

A G : G D = 2 : 1

The radius can be written as

A G = \frac{2}{3} A D

By substituting the values

A G = \frac{2}{3} \times \frac{9 3}{2}

So we get

A G - 33 c m

Therefore the radius of the circle is

33 c m

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Question Text	An equilateral triangle of side $9 c m$ is inscribed in a circle. Find the radius of the circle.
Answer Type	Text solution:1
Upvotes	153

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

Text solutionVerified

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An equilateral triangle of side $9 c m$ is inscribed in a circle. Find the radius of the circle.