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In triangle ABC, one angle is 60 degrees, the area is 10*sqrt(3) sq cm, and the perimeter is 20 cm. If a > b > c, where a, b, c are the sides opposite to angles A, B, C respectively, which statements are correct? (A) The inradius of the triangle equals sqrt(3). (B) The length of the longest side is 7. (C) The circumradius of the triangle equals 7/sqrt(3). (D) (incomplete)

1. (A) Inradius = sqrt(3)
2. (B) Longest side = 7
3. (C) Circumradius = 7/sqrt(3)
4. (D) (not available)

Correct answer: (A) Inradius = sqrt(3)

Solution

With angle C = 60 deg: area = (1/2)ab sin60 = 10*sqrt(3) gives ab = 40. Cosine rule gives c = 7. Then a+b = 13 and ab = 40, so a = 8, b = 5. Inradius r = Area/s = 10*sqrt(3)/10 = sqrt(3). Longest side = a = 8, not 7. Circumradius R = c/(2sinC) = 7/sqrt(3). So A and C are correct; B is false.

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