Question

If in $△ABC,a\cos A=b\cos B$, the $△ABC$ is

• A. Equilateral
• B. Isosceles
• C. Obtuse angled
• D. Isosceles or right angled

Answer 1

The correct option is D Isosceles or right angled

$\Rightarrow$ $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ [ Using sine law ]

$\Rightarrow$ $a=k\sin A,b=k\sin B,c=k\sin C$

Now, it is given that,

$\Rightarrow$ $a\cos A=b\cos B$

$\Rightarrow$ $k\sin A\cos A=k\sin B\cos B$

$\Rightarrow$ $\sin A\cos A=\sin B\cos B$

$\Rightarrow$ $2\sin A\cos A=2\sin B\cos B$

$\Rightarrow$ $\sin 2A=\sin 2B$

$\Rightarrow$ $\sin 2A-\sin 2B=0$

$\Rightarrow$ $2\sin (A-B)\cos (A+B)=0$

$\Rightarrow$ $\sin (A-B)\cos (\pi -C)=0$ [ As $A+B+C=\pi$ ]

$\Rightarrow$ $-\sin (A-B)\cos C=0$

$\Rightarrow$ $\cos C=0$ or $\sin (A-B)=0$

$\Rightarrow$ $C=\frac{\pi }{2}$ or $A-B=0$

$\Rightarrow$ $C=\frac{\pi }{2}$ or $A=B$

$\therefore$ The $△ABC,$ is right angle triangle or an isoceles triangle.