Question

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

Answer 1

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Radius = 5.19 cm

$\huge\star{\green{\underline{\mathfrak{Explanation :-}}}}$

Figure attached.

GIVEN :-

• AB=AC=BC= 9 cm
• The triangle is inscribed on a circle

TO FIND :-

• The radius of the circle OR AM

CONSTRUCTION :-

• AD as the median
• M is the centroid which actually divides thr median into a ratio of 2:1

SOLUTION :-

We know that,

AC= 9 cm and DC= 4.5(half of the radius)

So, applying Pythagoras theorem,

${AD}^{2}+{DC}^{2}={AC}^{2}$

Putting in the values,

${AD}^{2}+{(4.5)}^{2}={(9)}^{2}$

${AD}^{2}+20.25=81$

${AD}^{2}= 81-20.25$

${AD}^{2}=60.75$

$AD=\sqrt {60.75}$

$AD= 7.79 cm$

We also know that the centroid divides the median in the ratio of 2:1.

We need to find the the 2th part of the whole median.

We have total 3 parts.

So, $AM=\dfrac {2}{3}\times AD$

$AM =\dfrac {2}{3}\times 7.79$

$AM=\dfrac {\cancel {2}}{3}\times \dfrac{779}{\cancel{100}\:\: 50}$

$AM=\dfrac {779}{50\times 3}$

$AM=\dfrac {779}{150}$

$AM= 5.19 cm$

Hence, the radius = 5.19 cm

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