Question : $O$ is the circumcentre of the isosceles $\triangle ABC$. Given that $AB = AC = 5$ cm and $BC = 6$ cm. The radius of the circle is:
Option 1: 3.015 cm
Option 2: 3.205 cm
Option 3: 3.025 cm
Option 4: 3.125 cm
Correct Answer: 3.125 cm
Solution :
The circumradius $(R)$ of a triangle,
$ R = \frac{abc}{4K} $
where $a, b$ and $c$ are the sides of the triangle and $K$ is the area of the triangle.
Given that $AB = AC = 5$ cm ($a = b = 5$ cm) and $BC = 6$ cm ($c = 6$ cm).
The area $(K)$ of the triangle using Heron's formula,
$ K = \sqrt{s(s - a)(s - b)(s - c)} $ where $s$ is the semi-perimeter of the triangle.
$ s = \frac{a + b + c}{2} $
$ s = \frac{5 + 5 + 6}{2} = 8 $ cm
$ K = \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{8 \times 3 \times 3 \times 2} = 12 $ cm
$ R = \frac{5 \times 5 \times 6}{4 \times 12} = \frac{150}{48} = \frac{25}{8} = 3.125$ cm
Hence, the correct answer is 3.125 cm.
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