Statement:1 In `triangleABC`, if `a lt b sinA`, then the triangle is possible. And Statement:2 In `triangleABC a/(sinA)= b/(sinB)`

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-16
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-16
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the given problem, we will analyze the two statements regarding triangle ABC and determine their validity step by step.

### Step 1: Understand the Statements

1. **Statement 1**: In triangle ABC, if \( a < b \sin A \), then the triangle is possible.
2. **Statement 2**: In triangle ABC, \( \frac{a}{\sin A} = \frac{b}{\sin B} \).

### Step 2: Analyze Statement 2

According to the Sine Rule, which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides, we can express Statement 2 mathematically:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

This means that Statement 2 is indeed true as it directly follows from the Sine Rule.

### Step 3: Analyze Statement 1

Now, let's analyze Statement 1. We need to determine whether the condition \( a < b \sin A \) implies that triangle ABC is possible.

Using the Sine Rule, we know that:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

From this, we can derive that:

\[ a = \frac{b \sin A}{\sin B} \]

Now, if we assume \( a < b \sin A \), we can rewrite this as:

\[ \frac{b \sin A}{\sin B} < b \sin A \]

Dividing both sides by \( b \sin A \) (assuming \( b \sin A > 0 \)), we get:

\[ \frac{1}{\sin B} < 1 \]

This implies that \( \sin B > 1 \), which is not possible since the sine of an angle cannot exceed 1. Therefore, the condition \( a < b \sin A \) does not hold true for a valid triangle.

### Conclusion

- **Statement 1** is **false**.
- **Statement 2** is **true**.

### Final Answer

- Statement 1: False
- Statement 2: True

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