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Grade: 12th pass

                        

three sides of triangle are consecutive integers and the largest angle is twice the smallest angle.the perimeter of the triangle is...?

three sides of triangle are consecutive integers and the largest angle is twice the smallest angle.the perimeter of the triangle is...?

3 years ago

Answers : (1)

Arun
25768 Points
							
Please find the answer to your question

Let the sides of ?be n, n + 1, n + 2 where n? N.
Let a = n, b = n + 1, c = n + 2

235-1134_12345.png

 
Let the smallest, angle ? A = ? then the greatest ?C = 2? In ?ABC by applying Sine Law we get,
 
Sin ?/n = sin 2 ?/n + 2
 
? sin ?/n = 2 sin ? cos ?/ n + 2 ? 1/n = 2 cos ?/n + 2 (as sin ? ? 0)
 
? cos ? = n + 2/2n . . . . . . . . . . . . . . . . . . . . . . . . . (1)
 
In ? ABC by Cosine Law, we get
 
Cos ? = (n + 1)2 + (n + 2)2 – n2/2(n + 1) (n + 2) . . . . . . . . . . . . . . . . . . . (2)
 
Comparing the values of cos ? from (1) and (2), we get
 
(n + 1)2 + (n + 2)2 – n2/2(n + 1) (n + 2) = n + 2/2n
 
? (n + 2)2 (n + 1) = n(n + 2)2 + n(n + 1)2 – n3
 
? n (n + 2)2 (n + 2)2 = n(n + 2)2 + n(n + 1)2 – n3
 
? n2 + 4n + 4 = n3 + 2n2 + n – n3
 
? n2 – 3n – 4 = 0 ? (n + 1) (n – 4) = 0
 
? n = 4 (as n ? - 1)
 
? Sides of ? are 4, 4 + 1, 4 + 2, i.e. 4, 5, 6.
3 years ago
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