Question

The length of the sides of a right-angled triangle are in geometric progression. What is the ratio of the sines of its acute angles?

• A. $1$
• B. $\sqrt{3}$
• C. $\frac{\sqrt{5}+1}{2}$
• D. $\sqrt{\frac{\sqrt{5}+1}{2}}$

Answer 1

The correct option is D $\sqrt{\frac{\sqrt{5}+1}{2}}$

a,b,c are sides of Triangle

$b=ar,c={ar}^2$

Ratio of sines of acute angle $=\frac{\frac{b}{c}}{\frac{a}{c}}=\frac{b}{a}=r{c}^2{=a}^2{+b}^2{a}^2{r}^4=a^2+a^2{r}^2{r}^4-r^2-1=0r^2=\frac{1+\sqrt{5}}{2}r=\sqrt{\frac{\sqrt{5}+1}{2}}$

Ratio of sines of acute angle =$\sqrt{\frac{\sqrt{5}+1}{2}}$