If the sides of a triangle are in the ratio 9:6:4, then the triangle is said to be _____.

Options

A. right angled
B. obtuse angled
C. acute angled
D. All are incorrect

Answer

To determine the type of triangle based on the given side ratios of 9:6:4, we can follow these steps:

### Step 1: Identify the sides of the triangle
Given the ratio of the sides is 9:6:4, we can denote the sides as:
- Side A = 9x
- Side B = 6x
- Side C = 4x
where x is a common multiplier.

### Step 2: Identify the longest side
In this case, the longest side is 9x (Side A).

### Step 3: Apply the Pythagorean theorem for a right triangle
For a triangle to be a right triangle, it must satisfy the condition:
\[ A^2 + B^2 = C^2 \]
where C is the longest side. Here, we can check if:
\[ (6x)^2 + (4x)^2 = (9x)^2 \]

### Step 4: Calculate the squares
Calculating each side:
- \( (6x)^2 = 36x^2 \)
- \( (4x)^2 = 16x^2 \)
- \( (9x)^2 = 81x^2 \)

### Step 5: Add the squares of the two shorter sides
Now, add the squares of the two shorter sides:
\[ 36x^2 + 16x^2 = 52x^2 \]

### Step 6: Compare with the square of the longest side
Now, we compare this sum to the square of the longest side:
\[ 52x^2 \neq 81x^2 \]
Since \( 52x^2 \) is not equal to \( 81x^2 \), the triangle cannot be a right triangle.

### Step 7: Check for obtuse or acute triangle
For a triangle to be obtuse, the condition is:
\[ A^2 + B^2 < C^2 \]
We already calculated:
- \( A^2 + B^2 = 52x^2 \)
- \( C^2 = 81x^2 \)

Now check:
\[ 52x^2 < 81x^2 \]
This condition holds true, which means the triangle is obtuse.

### Conclusion
Since the triangle is not a right triangle and satisfies the condition for being obtuse, we conclude that the triangle is an obtuse triangle.

### Final Answer
The triangle is said to be **obtuse**.

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