If the sides of a triangle are proportional to the cosine of the opposite angles, then the triangle is ___________.
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Right angled
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equilateral
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obtuse angled
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None of these
By the Law of Cosines, cos(A) = (b^2 + c^2 - a^2) / (2bc). If sides are proportional to the cosines of opposite angles, a/cos(A) = b/cos(B) = c/cos(C). This condition is satisfied in an equilateral triangle where a=b=c and A=B=C=60 degrees.
Using the sine rule, the sides of a triangle are proportional to the sine of their opposite angles, giving a / sin A = b / sin B = c / sin C. If the sides are also proportional to the cosines of the opposite angles, we have sin A = cos A, sin B = cos B, and sin C = cos C for all angles. This implies tan A = 1, tan B = 1, and tan C = 1, meaning each angle equals 45 degrees. Since this makes the angle sum 135 degrees, which is impossible for a triangle, the only way a triangle can satisfy this is if it is equilateral, where all sides are equally proportional to each other. Therefore, the triangle is equilateral.