Suppose the sides of a triangle form a geomet | vedclass

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Question

(c)

Let the sides of triangle are

$a, a r, a r^2 . \quad[\because$ sides of triangle in $GP ]$

Case I $r > 1$

We know sum of two sides

Vedclass · Accepted Answer

$\left(\frac{1+\sqrt{5}}{2}, \frac{2+\sqrt{5}}{2}\right]$

Vedclass · Answer

(c) Let the sides of triangle are $a, a r, a r^2 . \quad[\because$ sides of triangle in $GP ]$ Case I $r > 1$ We know sum of two sides is greater than third side. $ herefore a+a r>a r^2 \Rightarrow r^2-r-1<0$ $\begin{array}{ll} \Rightarrow & r=\frac{1 \pm \sqrt{5}}{2} \\Rightarrow & 1 < r < \frac{\sqrt{5}+1}{2}, r > 1\end{array}$ Case $II$ $0 < r < 1$ $ herefore a r^2+a r > a \Rightarrow r^2+r-1 > 0$ $\Rightarrow \quad r=\frac{-1 \pm \sqrt{5}}{2}$ $\Rightarrow \quad 1 > r > \frac{\sqrt{5}-1}{2}, 0 < r < 1$ $ herefore \quad r \in\left(\frac{\sqrt{5}-1, \sqrt{5}+1}{2}, \frac{2}{2} ight)$

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval

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