The sides of a triangle are in the ratio 3:4:6. The triangle is :

Options

A. acute-angled
B. right-angled
C. obtuse-angled
D. either acute-angled or right angled

Answer

To determine the type of triangle formed by sides in the ratio 3:4:6, we can follow these steps:

### Step 1: Assign values to the sides of the triangle
Let the sides of the triangle be represented as:
- Side A = 3x
- Side B = 4x
- Side C = 6x
where x is a positive constant.

### Step 2: Identify the largest side
In the ratio 3:4:6, the largest side is 6x. We will denote this side as Side C.

### Step 3: Apply the triangle inequality condition
To classify the triangle, we need to check the relationship between the sides using the Pythagorean theorem:
- For a right triangle: \( c^2 = a^2 + b^2 \)
- For an acute triangle: \( c^2 < a^2 + b^2 \)
- For an obtuse triangle: \( c^2 > a^2 + b^2 \)

### Step 4: Calculate \( c^2 \), \( a^2 \), and \( b^2 \)
Now, we will calculate:
- \( c^2 = (6x)^2 = 36x^2 \)
- \( a^2 = (3x)^2 = 9x^2 \)
- \( b^2 = (4x)^2 = 16x^2 \)

### Step 5: Sum \( a^2 \) and \( b^2 \)
Now, we find the sum of \( a^2 \) and \( b^2 \):
- \( a^2 + b^2 = 9x^2 + 16x^2 = 25x^2 \)

### Step 6: Compare \( c^2 \) with \( a^2 + b^2 \)
Now we compare:
- \( c^2 = 36x^2 \)
- \( a^2 + b^2 = 25x^2 \)

Since \( 36x^2 > 25x^2 \), we conclude that:
- \( c^2 > a^2 + b^2 \)

### Step 7: Conclusion
According to the conditions we established:
- Since \( c^2 > a^2 + b^2 \), the triangle is classified as an **obtuse triangle**.