Question
In ∆ ABC, ∠ A is the largest angle and ∠ B is
the smallest angle. If the length of the sides of the triangle are consecutive positive integers, then the length of the largest side (in units) is:Solution
According to the question, ∠ A = 2∠ B and the length of the sides of the triangle are consecutive positive integers. So, a = x + 1 c = x b = x - 1 Now, by using the sine rule. (sin 2B)/ (x + 1) = (sin B)/ (x - 1) (sin 2B)/ (x + 1) =sinB/ (x-1) 2sinBcosB/ x + 1 = (sin B)/ (x - 1) 2cosB= (x + 1)/ (x - 1) … (1) Now, by using the cosine rule, cosB= (a2+c2-b2)/2ac On substituting the value of the respective variable in the above formula, we get, cosB=(x+1)2 +x2-(x-1)2/ 2(x+1) x cosB=(x2+1+2x+x2-x2-1+2x)/ 2x2+2x cosB=(4+x)/2(x+1) … (2) On equating (1) and (2), we get, (4+x)/(x+1) =(x+1)/(x-1) = 4x-4 + x2 - x = x² + 1 + 2x x=5 units The length of the largest side, x+1=5+1 =6 units
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