If the angles of a triangle are in the ratio 1 : 2 : 7, then the ratio of the greatest side to the least side, is

# If the angles of a triangle are in the ratio 1 : 2 : 7, then the ratio of the greatest side to the least side, is

1. A

$\left(\sqrt{5}-1\right):\left(\sqrt{5}+1\right)$

2. B

$\left(\sqrt{5}+1\right):\left(\sqrt{5}-1\right)$

3. C

$\left(\sqrt{5}+2\right):\left(\sqrt{5}-2\right)$

4. D

$\left(\sqrt{5}-2\right):\left(\sqrt{5}+2\right)$

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### Solution:

Let the angles of triangle ABC be  and $C=7\theta .$ Then,

Clearly, c is the greatest side and a is the smallest side.

Now,

$\begin{array}{l}\frac{a}{\mathrm{sin}A}=\frac{c}{\mathrm{sin}C}\\ ⇒\frac{c}{a}=\frac{\mathrm{sin}C}{\mathrm{sin}A}=\frac{\mathrm{sin}{126}^{\circ }}{\mathrm{sin}{18}^{\circ }}=\frac{\mathrm{cos}{36}^{\circ }}{\mathrm{sin}{18}^{\circ }}=\frac{\sqrt{5}+1}{\sqrt{5}-1}\end{array}$

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