MHT CET · Maths · Properties of Triangles
With usual notations, in any \(\triangle A B C\), if a \(\cos B=b \cos A\), then the triangle
- A an isosceles triangle
- B an equilateral triangle
- C a right angled triangle
- D a scalene triangle
Answer & Solution
Correct Answer
(A) an isosceles triangle
Step-by-step Solution
Detailed explanation
We know that \(\frac{a}{\sin A}=\frac{b}{\sin B}\)
\(\therefore\text a \sin\text B=\text b \sin\text A \quad\ldots(1)\) and we have
\(\text a \cos\text B=\text b \cos\text A\quad\ldots(2)\)
From (1) and (2), we write
\(
\frac{\mathrm{a}}{\mathrm{b}}=\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\cos \mathrm{A}}{\cos \mathrm{B}} \Rightarrow \tan \mathrm{A}=\tan \mathrm{B} \Rightarrow \mathrm{A}=\mathrm{B}
\)
Thus \(\Delta\) is an isosceles triangle.
\(\therefore\text a \sin\text B=\text b \sin\text A \quad\ldots(1)\) and we have
\(\text a \cos\text B=\text b \cos\text A\quad\ldots(2)\)
From (1) and (2), we write
\(
\frac{\mathrm{a}}{\mathrm{b}}=\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\cos \mathrm{A}}{\cos \mathrm{B}} \Rightarrow \tan \mathrm{A}=\tan \mathrm{B} \Rightarrow \mathrm{A}=\mathrm{B}
\)
Thus \(\Delta\) is an isosceles triangle.
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