Question

If the sides of a triangle are in A.P. and the greatest angle of the triangle is double the smallest angle, then the ratio of the sides of the triangle is

• A. 3:4:5
  
• B. 4:5:6
  
• C. 5:6:7
  
• D. 7:8:9

Answer 1

The correct option is B

4:5:6

Let the sides of the triangle be a-d,a and a+d with a>d>0. Clearly, a-d is the smallest side and a+d is the largest side. So A is the smallest and C is the largest angle. Then c=2A. Thus the angles of the triangle are, Applying the law of sines we have
$\frac{a-d}{sinA}=\frac{a}{sin(\pi -3A)}=\frac{a+d}{sin2A}$
$\Rightarrow \frac{a-d}{sinA}=\frac{a}{sin3A}=\frac{a+d}{sin2A}\Rightarrow \frac{a-d}{sinA}=\frac{a}{3sinA-4si{n}^{3}A}=\frac{a+d}{2sinAcosA}$
$\Rightarrow \frac{a-d}{1}=\frac{a}{3-4si{n}^{2}A}=\frac{a+d}{2cosA}\Rightarrow 3-4si{n}^{2}A=\frac{a}{a-d}$
2cosA $=\frac{a+d}{a-d}$
$\Rightarrow 4co{s}^{2}A-1=\frac{a}{a-d}$
$2cosA=\frac{a+d}{a-d}$
$\Rightarrow {\left(\frac{a+d}{a-d}\right)}^2-1=\frac{a}{a-d}$
$\Rightarrow a=5d$
Thus, the sides of the triangle are 4d, 5d, 6d. Hence, their ratio is 4:5:6