Question

In a triangle the cosines of two angles are inversely proportional to the sides opposite the angles.

The triangle is?

• A. isosceles
• B. equilateral
• C. right angled
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A isosceles
C right angled

From given question,
$\frac{\cos B}{\cos A}=\frac{a}{b}$
$\frac{a}{\cos B}=\frac{b}{\cos A}$
$\frac{a}{\sin ({90}^0-B)}=\frac{b}{\sin ({90}^0-A)}$
Following sine rule, we get
${90}^0-B=A$
$A+B={90}^0$
${180}^0-C={90}^0$
$C={90}^0$ ....right angled triangle.
Now if $a=b$, then $A=B$ ... isosceles.