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
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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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