Question

Suppose the sides of a triangle from a geometric progression with common ratio r. Then r Iies in the interval

• A. $(0,\frac{-1+\sqrt{5}}{2})$
• B. $(\frac{1+\sqrt{5}}{2},\frac{2+\sqrt{5}}{2})$
• C. $(\frac{-1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})$
• D. $(\frac{2+\sqrt{5}}{2},\infty )$

Answer 1

The correct option is C $(\frac{-1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})$

Let the sides as $a,ar,a{r}^2$

$a+b>ca+ar>a{r}^{2}a{r}^2-ar-a<0(r^2-r-1)<0r=\frac{1\pm \sqrt{1+4}}{2}r\in (\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})$