Question

The lengths 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. $\frac{\sqrt{5} - 1}{2}$

Hint:

In GP triangle problems, assign $a, ar, ar^2$, apply Pythagoras, and solve for $r$ systematically.

Answer 1

The Correct Option is C

Let sides in GP be $a, ar, ar^2$. Hypotenuse is largest: $ar^2$. Pythagoras: \[ a^2 + a^2 r^2 = a^2 r^4 \] Divide by $a^2$: \[ 1 + r^2 = r^4 \quad \Rightarrow \quad r^4 - r^2 - 1 = 0 \] Let $u = r^2$: \[ u^2 - u - 1 = 0 \quad \Rightarrow \quad u = \frac{1 + \sqrt{5}}{2} = \phi \] Thus: \[ r = \sqrt{\phi} \] Now: $\sin(\theta_1) = \frac{a}{ar^2} = \frac{1}{r^2} = \frac{1}{\phi}$, $\sin(\theta_2) = \frac{ar}{ar^2} = \frac{1}{r} = \frac{1}{\sqrt{\phi}}$. Ratio: \[ \frac{\sin \theta_2}{\sin \theta_1} = \frac{1/\sqrt{\phi}}{1/\phi} = \sqrt{\phi} = \frac{\sqrt{5} + 1}{2} \]

\[ \boxed{\frac{\sqrt{5} + 1}{2}} \]