Question

In a triangle ABC, if $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then triangle ABC will be:

• A. Isosceles triangle
• B. Equilateral triangle
• C. Right triangle
• D. Right Isosceles triangle

Answer 1

Answer: (A), (B)

Equilateral triangle

We are given that,
$\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\Rightarrow \frac{\cos A}{2R\sin A}=\frac{\cos B}{2R\sin B}=\frac{\cos C}{2R\sin C}[\because \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R]\Rightarrow \tan A=\tan B=\tan C\Rightarrow \angle A=\angle B=\angle C\mathrm{Hence},\mathrm{ABC}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}.$
Every equilateral triangle is an isosceles triangle.