Question

In $\Delta ABC,$ if $A:B:C=1:5:6$ then $a:b:c=$

• A. 1) $\left(\sqrt{3}-1\right):2\sqrt{2}:\left(\sqrt{3}+1\right)$
• B. 2) $2\sqrt{2}:\left(\sqrt{3}-1\right):\left(\sqrt{3}+1\right)$
• C. 3) $\left(\sqrt{3}-1\right):\left(\sqrt{3}+1\right):2\sqrt{2}$
• D. 4) $\left(\sqrt{3}-1\right):\sqrt{3}:\sqrt{2}$

Answer 1

Answer: (B)

2) $2\sqrt{2}:\left(\sqrt{3}-1\right):\left(\sqrt{3}+1\right)$

Given:$A:B:C=1:5:6$

Let $A=x,B=5x,C=6x$

$\Rightarrow A+B+C={180}^{\circ }$

$\Rightarrow x+5x+6x={180}^{\circ }$

$\Rightarrow 12x={180}^{\circ }$

$\Rightarrow x=\frac{{180}^{\circ }}{12}={15}^{\circ }$

$\Rightarrow A=x={15}^{\circ },B=5x=5\times {15}^{\circ }={75}^{\circ },C=6\times {15}^{\circ }={90}^{\circ }$

$\Rightarrow A={15}^{\circ },B={75}^{\circ },C={90}^{\circ }$

Using Sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

$\Rightarrow \frac{a}{\sin {15}^{\circ }}=\frac{b}{\sin {75}^{\circ }}=\frac{c}{\sin {90}^{\circ }}=2R$

We know that $\sin {15}^{\circ }=\frac{\sqrt{3}-1}{2\sqrt{2}},\sin {75}^{\circ }=\frac{\sqrt{3}+1}{2\sqrt{2}}$ and $\sin {90}^{\circ }=1$

$\Rightarrow \frac{a}{\frac{\sqrt{3}-1}{2\sqrt{2}}}=\frac{b}{\frac{\sqrt{3}+1}{2\sqrt{2}}}=\frac{c}{1}=2R$

$\Rightarrow \frac{a}{\frac{\sqrt{3}-1}{2\sqrt{2}}}=2R,\frac{b}{\frac{\sqrt{3}+1}{2\sqrt{2}}}=2R,\frac{c}{1}=2R$

$\Rightarrow a=2R\frac{\sqrt{3}-1}{2\sqrt{2}},b=2R\frac{\sqrt{3}+1}{2\sqrt{2}},c=2R$

$\Rightarrow a:b:c=2R\frac{\sqrt{3}-1}{2\sqrt{2}}:2R\frac{\sqrt{3}+1}{2\sqrt{2}}:2R$

or $a:b:c=\frac{\sqrt{3}-1}{2\sqrt{2}}:\frac{\sqrt{3}+1}{2\sqrt{2}}:1$

or $a:b:c=\sqrt{3}-1:\sqrt{3}+1:2\sqrt{2}$