If in a `DeltaABC` ,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 \( \Delta ABC \): \( A \sin A = B \sin B \).

### Step-by-step Solution:

1. **Understand the Given Condition**:
We start with the equation \( A \sin A = B \sin B \). This implies a relationship between the sides \( A \) and \( B \) of triangle \( ABC \).

**Hint**: Recall the sine rule, which relates the sides of a triangle to the sines of its angles.

2. **Use 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.

**Hint**: This means that each side divided by the sine of its opposite angle is equal to a constant.

3. **Express \( A \sin A \) and \( B \sin B \)**:
From the sine rule, we can express \( A \) and \( B \) in terms of \( k \):
\[ A = k \sin A \quad \text{and} \quad B = k \sin B \]

**Hint**: Substitute these expressions back into the original equation.

4. **Substituting into the Equation**:
Substitute \( A \) and \( B \) into the equation \( A \sin A = B \sin B \):
\[ (k \sin A) \sin A = (k \sin B) \sin B \]
This simplifies to:
\[ k \sin^2 A = k \sin^2 B \]

**Hint**: Since \( k \) is not zero (as it represents a ratio of sides), we can divide both sides by \( k \).

5. **Canceling \( k \)**:
Dividing both sides by \( k \) gives us:
\[ \sin^2 A = \sin^2 B \]

**Hint**: Recall the properties of sine functions and their implications on angles.

6. **Conclusion about Angles**:
From \( \sin^2 A = \sin^2 B \), we conclude that:
\[ \sin A = \sin B \quad \text{or} \quad \sin A = -\sin B \]
Since angles in a triangle cannot be negative, we have:
\[ A = B \]

**Hint**: If two angles in a triangle are equal, then the sides opposite those angles are also equal.

7. **Final Result**:
Since \( A = B \), it follows that the triangle \( \Delta ABC \) is an isosceles triangle.

**Hint**: An isosceles triangle has at least two equal sides.

### Final Answer:
The triangle \( \Delta ABC \) is an **isosceles triangle**.