Question

If the sides of a right angled triangle are in GP, then cosine of the greater acute angle is

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

Let the sides of the right angled triangle be a ar, $a{r}^2$ of which $a{r}^2$ is the hypotenuse, then r > 1.
Now, $a^2{r}^4$ = a^2 + a^2r^2\) or
$r^4-r^2-1=0$ $\therefore r^2=\frac{1\pm \sqrt{5}}{2}$
$\because$ r>1 $\therefore$ $r^2$ >1 $\therefore r^2=\frac{1+\sqrt{5}}{2}$
Angle C is the greater acute angle.
Now, cos $C=\frac{a}{a{r}^2}$=$\frac{1}{r^2}=\frac{2}{1+\sqrt{5}}$