Properties of Triangles 1 Question 20
20. The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.
(1991, 4M)
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Solution:
- Let $A B C$ be the triangle such that the lengths of its sides $C A, A B$ and $B C$ are $(x-1), x$ and $(x+1)$ respectively, where $x \in N$ and $x>1$. Let $\angle B=\alpha$ be the smallest angle and $\angle A=2 \alpha$ be the largest angle.
Then, by sine rule, we have
$$ \begin{array}{rlrl} & & \frac{\sin \alpha}{x-1} & =\frac{\sin 2 \alpha}{x+1} \\ \Rightarrow & & \frac{\sin 2 \alpha}{\sin \alpha} & =\frac{x+1}{x-1} \\ \Rightarrow & 2 \cos \alpha & =\frac{x+1}{x-1} \\ \therefore & \cos \alpha & =\frac{x+1}{2(x-1)} \end{array} $$
Also, $\cos \alpha=\frac{x^{2}+(x+1)^{2}-(x-1)^{2}}{2 x(x+1)} \quad$ [using cosine law]
$$ \Rightarrow \quad \cos \alpha=\frac{x+4}{2(x+1)} $$
From Eqs. (i) and (ii),
$$ \begin{aligned} & & \frac{x+1}{2(x-1)} & =\frac{x+4}{2(x+1)} \\ \Rightarrow & & (x+1)^{2} & =(x+4)(x-1) \\ \Rightarrow & & x^{2}+2 x+1 & =x^{2}+3 x-4 \\ \Rightarrow & & x & =5 \end{aligned} $$
Hence, the lengths of the sides of the triangle are 4, 5 and 6 units.
