If in a `triangle ABC` ,a sin A=b sin B, then the triangle, is

Options

A. right angled
B. equilaterial
C. isosceles
D. none of these

Answer

To solve the problem, we need to analyze the given condition in triangle ABC, which states that \( a \sin A = b \sin B \). We will derive the conclusion step by step.

### Step-by-Step Solution:

1. **Given Condition**:
We start with the equation given in the problem:
\[ a \sin A = b \sin B \]

2. **Using the Sine Rule**:
According to the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = K \]
where \( K \) is a constant.

3. **Expressing Sine Values**:
From the sine rule, we can express \( \sin A \) and \( \sin B \) in terms of \( K \):
\[ \sin A = \frac{a}{K} \quad \text{and} \quad \sin B = \frac{b}{K} \]

4. **Substituting into the Given Condition**:
Substitute these expressions into the original equation:
\[ a \left(\frac{a}{K}\right) = b \left(\frac{b}{K}\right) \]
This simplifies to:
\[ \frac{a^2}{K} = \frac{b^2}{K} \]

5. **Canceling K**:
Since \( K \) is a constant and cannot be zero, we can cancel \( K \) from both sides:
\[ a^2 = b^2 \]

6. **Taking the Square Root**:
Taking the square root of both sides gives us:
\[ a = \pm b \]
Since side lengths cannot be negative, we discard the negative solution:
\[ a = b \]

7. **Conclusion**:
If \( a = b \), it means that triangle ABC has two equal sides. Therefore, triangle ABC is an **isosceles triangle**.

### Final Answer:
The triangle is **isosceles**.

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