ΔABC is inscribed in a circle with center O. If AB = 17 cm, BC = 10 cm and AC = 9 cm, then find the

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ΔABC is inscribed in a circle with center O. If AB = 17 cm, BC = 10 cm and AC = 9 cm, then find the length of AO.

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Find Semiperimeter):","type":"text"},{"text":" The side lengths of ","type":"text"},{"attrs":{"latex":"\triangle ABC"},"type":"math_inline"},{"text":" are ","type":"text"},{"attrs":{"latex":"a = 10"},"type":"math_inline"},{"text":" cm, ","type":"text"},{"attrs":{"latex":"b = 9"},"type":"math_inline"},{"type":"text","text":" cm, and "},{"attrs":{"latex":"c = 17"},"type":"math_inline"},{"text":" cm. We find the semiperimeter ","type":"text"},{"attrs":{"latex":"s"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"s = \frac{a + b + c}{2} = \frac{10 + 9 + 17}{2} = \frac{36}{2} = 18 \text{ cm}"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Calculate Triangle Area using Heron's):","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}"},"type":"math_block"},{"attrs":{"latex":"\text{Area} = \sqrt{18(18-10)(18-9)(18-17)}"},"type":"math_block"},{"type":"math_block","attrs":{"latex":"\text{Area} = \sqrt{18 \times 8 \times 9 \times 1}"}},{"attrs":{"latex":"\text{Area} = \sqrt{1296} = 36 \text{ sq. cm}"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Apply Circumradius Formula):","type":"text"},{"text":" The radius ","type":"text"},{"type":"math_inline","attrs":{"latex":"R"}},{"text":" of the circumcircle (which equals the length of ","type":"text"},{"attrs":{"latex":"AO"},"type":"math_inline"},{"type":"text","text":", where "},{"attrs":{"latex":"O"},"type":"math_inline"},{"type":"text","text":" is the center) is given by:"}],"type":"paragraph"},{"attrs":{"latex":"R = \frac{abc}{4 \times \text{Area}}"},"type":"math_block"},{"attrs":{"latex":"R = \frac{10 \times 9 \times 17}{4 \times 36}"},"type":"math_block"},{"attrs":{"latex":"R = \frac{1530}{144} = \frac{85}{8} = 10.625 \text{ cm}"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"text":" The length of ","type":"text"},{"attrs":{"latex":"AO"},"type":"math_inline"},{"text":" is exactly ","type":"text"},{"text":"10.625","type":"text","marks":[{"type":"code"}]},{"text":" cm (or ","type":"text"},{"marks":[{"type":"code"}],"text":"85/8","type":"text"},{"text":").","type":"text"}],"type":"paragraph"}],"type":"doc"}

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