Trigonometric Equations Question 76
Question: The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are
Options:
A) 1, 2, 3
B) 2, 3, 4
C) 3, 4, 5
D) 4, 5, 6
Show Answer
Answer:
Correct Answer: D
Solution:
- Let the sides of $ \Delta ABC $ be $ a=n,b=n+1,c=n+2 $ , where n is a natural number. Then C is the greatest and A the least angle. As given $ C=2A $ . \ $ \sin C=\sin 2A=2\sin A\cos A $ \ $ kc=2ka\frac{b^{2}+c^{2}-a^{2}}{2bc} $ or $ bc^{2}=a(b^{2}+c^{2}-a^{2}) $ Substituting the values of a, b, c, we get $ (n+1){{(n+2)}^{2}}=n[{{(n+1)}^{2}}+{{(n+2)}^{2}}-n^{2}] $ or $ (n+1){{(n+2)}^{2}}=n(n^{2}+6n+5)=n(n+1)(n+5) $ Since $ n\ne -1 $ , we can cancel $ n+1 $ . Thus $ {{(n+2)}^{2}}=n(n+5) $ or $ n^{2}+4n+4=n^{2}+5n $ This gives $ n=4, $ Hence the sides are 4, 5 and 6.
BETA
