Question

Angles $A,B,C$ of a $∆ABC$ are in Arithmetic progression & $b:c=\sqrt{3}:\sqrt{2}$, then $\angle A$

• A. ${45}^{°}$
• B. ${60}^{°}$
• C. ${75}^{°}$
• D. ${90}^{°}$

Answer 1

The correct option is C

${75}^{°}$

Explanation for the correct option:

Step 1. Find the value of angle $A:$

Given, angles of a triangle $ABC$ are in $AP.$

Let $a-d,a,a+d$ represent angles $A,B$ and $C.$

We know that $\mathit{A}\mathbf{+}\mathit{B}\mathbf{+}\mathit{C}\mathbf{=}{\mathbf{180}}^{\mathbf{\circ }}$

$a-d+a+a+d$$=180$ $[\because$sum of angles of triangle$={180}^{\circ }$$]$

$\Rightarrow$ $3a-d+d=180$

$\Rightarrow$ $3a$$=180$

$\Rightarrow$ $a=\frac{180}{3}$

$\Rightarrow$ $a=60$

So $\angle B$$={60}^{\circ }$

Also Given, $b:c$$=\sqrt{3}:\sqrt{2}$

By sine rule, We know that, If $A,B$ and $C$are angles and, $a,b$ and $c$ are the sides of $∆ABC$, then:

$\left(\frac{a}{\sin \left(A\right)}=\frac{b}{\sin \left(B\right)}=\frac{c}{\sin \left(C\right)}\right)$

$\left(\frac{b}{c}=\frac{\sin \left(B\right)}{\sin \left(C\right)}\right)$ ….(1)

$\begin{array}{rcl}\sin \left(B\right)& =& \sin \left({60}^{\circ }\right)\\ & =& \frac{\sqrt{3}}{2}\end{array}$

Step 2. Put the value of $b,c,\sin \left(B\right)$ in equation (1), we get

$\frac{\sqrt{3}}{\sqrt{2}}=\frac{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}{\sin \left(C\right)}$

$\Rightarrow$ $\frac{\sqrt{3}}{\sqrt{2}}\times \frac{2}{\sqrt{3}}=\frac{1}{\sin \left(C\right)}$

$\Rightarrow$ $\frac{\sqrt{3}\times 2}{\sqrt{2}\times \sqrt{3}}=\frac{1}{\sin \left(C\right)}$

$\Rightarrow$ $\frac{\sqrt{3}\times \sqrt{2}\times \sqrt{2}}{\sqrt{2}\times \sqrt{3}}=\frac{1}{\sin \left(C\right)}$

$\Rightarrow$ $\sqrt{2}=\frac{1}{\sin \left(C\right)}$

$\Rightarrow$ $\sin \left(C\right)\mathbf{=}\frac{1}{\sqrt{2}}$

$\Rightarrow$ $C={\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

So $C={45}^{\circ }$

Step 3. Put the values of angle $B,C$ in $\mathit{A}\mathbf{+}\mathbf{}\mathit{B}\mathbf{+}\mathit{C}\mathbf{=}{\mathbf{180}}^{\mathbf{\circ }}$

$\Rightarrow$ $A+{60}^{\circ }+{45}^{\circ }={180}^{\circ }$

$\Rightarrow$ $\mathit{A}$$\mathbf{=}{\mathbf{75}}^{\mathbf{\circ }}$

Hence, Option ‘C' is correct Answer.