If the data given to construct a triangle ABC are a = 5, b= 7, sin `A=3//4`, then it is possible to construct

Options

A. only one triangle
B. two triangles
C. infinitely many triangles
D. no triangles

Answer

To determine whether it is possible to construct triangle ABC with the given data \( a = 5 \), \( b = 7 \), and \( \sin A = \frac{3}{4} \), we can follow these steps:

### Step 1: Use the Sine Rule
The Sine Rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Given \( a = 5 \), \( b = 7 \), and \( \sin A = \frac{3}{4} \), we can substitute these values into the Sine Rule.

### Step 2: Substitute the Values
Substituting the known values into the Sine Rule gives:
\[ \frac{5}{\frac{3}{4}} = \frac{7}{\sin B} \]

### Step 3: Simplify the Equation
To simplify the left side:
\[ \frac{5}{\frac{3}{4}} = 5 \cdot \frac{4}{3} = \frac{20}{3} \]
Thus, the equation becomes:
\[ \frac{20}{3} = \frac{7}{\sin B} \]

### Step 4: Cross-Multiply
Cross-multiplying gives:
\[ 20 \sin B = 21 \]

### Step 5: Solve for \( \sin B \)
Now, we can solve for \( \sin B \):
\[ \sin B = \frac{21}{20} \]

### Step 6: Analyze the Result
Since the value of \( \sin B \) must be in the range of \([-1, 1]\), and \( \frac{21}{20} = 1.05\) which is greater than 1, it is not possible for \( \sin B \) to have this value.

### Conclusion
Therefore, it is not possible to construct triangle ABC with the given data.

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