Which of the following is the CORRECT option for the triangles having sides in the ratio of 3:4:6?

Options

A. Acute angled
B. Obtuse angled
C. Right angled
D. Either acute right angled

Answer

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

### Step 1: Assign the sides
Let the sides of the triangle be represented as:
- Side a = 3k
- Side b = 4k
- Side c = 6k
where k is a positive constant.

### Step 2: Calculate the squares of the sides
Now, we will calculate the squares of each side:
- a² = (3k)² = 9k²
- b² = (4k)² = 16k²
- c² = (6k)² = 36k²

### Step 3: Check the triangle inequality
To determine the type of triangle, we will check the triangle inequality theorem:
- The sum of the squares of the two shorter sides (a and b) should be compared with the square of the longest side (c).

Calculate the sum of the squares of the two shorter sides:
- a² + b² = 9k² + 16k² = 25k²

Now compare this with c²:
- c² = 36k²

### Step 4: Analyze the comparison
Now we compare:
- 25k² < 36k²

Since the sum of the squares of the two shorter sides is less than the square of the longest side, this indicates that the triangle is an obtuse triangle.

### Conclusion
Thus, the correct option for the triangles having sides in the ratio of 3:4:6 is that it forms an obtuse triangle.

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