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Properties of Triangle
Properties of Triangle
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Question

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x 2 – c 2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

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Solution

Given a + b = x and ab = y If x 2 −- c 2 = y ⇒\Rightarrow (a + b) 2 −- c 2 = ab ⇒\Rightarrow a 2 + b 2 −- c 2 = −- ab ⇒\Rightarrow a2+b2−c22ab=−12{{{a^2} + {b^2} - {c^2}} \over {2ab}} = - {1 \over 2} ⇒cos⁡C=−12 \Rightarrow \cos C = - {1 \over 2} ⇒∠C=2π3 \Rightarrow \angle C = {{2\pi } \over 3} R=c2sin⁡C=c3R = {c \over {2\sin C}} = {c \over {\sqrt 3 }}

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