Question

With usual notations, in any $\Delta ABC$, if $a\cos B = b\cos A$, then the triangle is

• A. an isosceles triangle
• B. an equilateral triangle
• C. a right angled triangle
• D. a scalene triangle

Hint:

You can use substitution to test simple special cases. An equilateral triangle has $A=B=60^\circ$, which satisfies the condition since $\cos 60^\circ = \cos 60^\circ$. However, an equilateral triangle is a special case of an isosceles triangle. Testing a case where $A \neq B$ will show it fails, confirming the broader classification is an isosceles triangle.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a geometric triangle identity constraint: $a\cos B = b\cos A$. We need to identify the specific classification of triangle $ABC$ that always satisfies this condition.

Step 2: Key Formula or Approach:
According to the Sine Rule for triangles, the side lengths are directly proportional to the sines of their opposite angles: $$\frac{a}{\sin A} = \frac{b}{\sin B} \implies a = k\sin A, \quad b = k\sin B$$ We will substitute these expressions into the given relation to convert it into a purely trigonometric equation.

Step 3: Detailed Explanation:
Substitute $a = k\sin A$ and $b = k\sin B$ into the given condition $a\cos B = b\cos A$: $$(k\sin A)\cos B = (k\sin B)\cos A$$ Since $k \neq 0$, we can divide both sides by $k$: $$\sin A\cos B = \sin B\cos A$$ Rearranging the terms onto the left side: $$\sin A\cos B - \cos A\sin B = 0$$ Recognize this as the standard sine difference identity $\sin(A-B) = \sin A\cos B - \cos A\sin B$: $$\sin(A - B) = 0$$ Within any triangle, the internal angles $A$ and $B$ are strictly bounded between $0^\circ$ and $180^\circ$, which means the difference range is $-180^\circ \lt A - B \lt 180^\circ$. The only value where the sine function vanishes in this domain is 0: $$A - B = 0 \implies A = B$$ Since two internal angles of the triangle are equal, the sides opposite to those angles must also be equal ($a = b$). This satisfies the definition of an isosceles triangle.

Step 4: Final Answer:
The triangle is an isosceles triangle, which corresponds to option (A).