Question

If in a $\Delta ABC$, $\cos A+\cos B+\cos C=\frac{3}{2}$, then triangle is

• A. Right angled
• B. Isosceles
• C. Acute
• D. Equilateral

Answer 1

The correct option is D Equilateral

Given, $\cos A+\cos B+\cos C=\frac{3}{2}$
$\Rightarrow \frac{b^2+c^2-a^2}{2bc}+\frac{a^2+c^2-b^2}{2ac}+\frac{a^2+b^2-c^2}{2ab}=\frac{3}{2}$
$\Rightarrow (a+b-c){(a-b)}^2+(b+c-a){(b-c)}^2+(c+a-b){(c-a)}^2=0$
As we know,
$a+b>c$, $b+c>a$, $c+a>b$
$\Rightarrow a=b=c$

Hence, the triangle is an equilateral.