If in a `Delta`ABC, cos B = (sin A)/(2 sin C), then the triangle is

Options

A. Isosceles triangle
B. Equilateral triangle
C. Right angled triangle
D. Scalene triangle

Explanation

To solve the problem, we need to analyze the given equation in the context of triangle properties. Let's go through the steps systematically.

### Step 1: Understand the Given Condition
We are given that:
\[ \cos B = \frac{\sin A}{2 \sin C} \]
This is our starting point.

**Hint:** Recall the relationship between the sides and angles in a triangle, particularly the Law of Cosines and the Law of Sines.

### Step 2: Apply the Law of Cosines
Using the Law of Cosines, we can express \(\cos B\) in terms of the sides of the triangle:
\[ \cos B = \frac{A^2 + C^2 - B^2}{2AC} \]
where \(A\), \(B\), and \(C\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively.

**Hint:** The Law of Cosines relates the angles of a triangle to the lengths of its sides.

### Step 3: Apply the Law of Sines
Using the Law of Sines, we have:
\[ \frac{A}{\sin A} = \frac{C}{\sin C} \]
From this, we can express \(\sin A\) in terms of \(A\) and \(C\):
\[ \sin A = \frac{A \cdot \sin C}{C} \]

**Hint:** The Law of Sines allows us to relate the angles and sides of a triangle.

### Step 4: Substitute \(\sin A\) into the Given Condition
Now substitute \(\sin A\) into the equation:
\[ \cos B = \frac{\frac{A \cdot \sin C}{C}}{2 \sin C} = \frac{A}{2C} \]

**Hint:** Simplifying the equation will help us relate the sides of the triangle.

### Step 5: Set Up the Equation
Now we have:
\[ \frac{A^2 + C^2 - B^2}{2AC} = \frac{A}{2C} \]
Cross-multiplying gives:
\[ A^2 + C^2 - B^2 = A \cdot \frac{AC}{C} = A^2 \]
This simplifies to:
\[ C^2 - B^2 = 0 \]

**Hint:** This indicates a relationship between the sides of the triangle.

### Step 6: Solve for the Relationship Between Sides
From \(C^2 - B^2 = 0\), we can conclude that:
\[ C^2 = B^2 \implies C = B \]
This means that two sides of the triangle are equal.

**Hint:** When two sides of a triangle are equal, it indicates that the triangle is isosceles.

### Conclusion
Since we have established that \(B = C\), we conclude that triangle \(ABC\) is an **isosceles triangle**.

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