Which of the following ratios can be the ratio of the sides of a right angled triangle ?

Options

A. `9:6:3`
B. `13:12:5`
C. `7:6:5`
D. `5:3:2`

Answer

To determine which of the given ratios can represent the sides of a right-angled triangle, we will use the Pythagorean theorem. According to this theorem, in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Let's analyze each option step by step:

### Step 1: Analyze Option A - 9:6:3
1. Assign the sides based on the ratio:
- Let the sides be 9k, 6k, and 3k.
2. Identify the longest side (hypotenuse):
- The longest side is 9k.
3. Apply the Pythagorean theorem:
- \( (9k)^2 = (6k)^2 + (3k)^2 \)
- \( 81k^2 = 36k^2 + 9k^2 \)
- \( 81k^2 = 45k^2 \)
4. This is not true, so this ratio does not represent a right-angled triangle.

### Step 2: Analyze Option B - 13:12:5
1. Assign the sides based on the ratio:
- Let the sides be 13k, 12k, and 5k.
2. Identify the longest side (hypotenuse):
- The longest side is 13k.
3. Apply the Pythagorean theorem:
- \( (13k)^2 = (12k)^2 + (5k)^2 \)
- \( 169k^2 = 144k^2 + 25k^2 \)
- \( 169k^2 = 169k^2 \)
4. This is true, so this ratio can represent a right-angled triangle.

### Step 3: Analyze Option C - 7:6:5
1. Assign the sides based on the ratio:
- Let the sides be 7k, 6k, and 5k.
2. Identify the longest side (hypotenuse):
- The longest side is 7k.
3. Apply the Pythagorean theorem:
- \( (7k)^2 = (6k)^2 + (5k)^2 \)
- \( 49k^2 = 36k^2 + 25k^2 \)
- \( 49k^2 = 61k^2 \)
4. This is not true, so this ratio does not represent a right-angled triangle.

### Step 4: Analyze Option D - 5:3:2
1. Assign the sides based on the ratio:
- Let the sides be 5k, 3k, and 2k.
2. Identify the longest side (hypotenuse):
- The longest side is 5k.
3. Apply the Pythagorean theorem:
- \( (5k)^2 = (3k)^2 + (2k)^2 \)
- \( 25k^2 = 9k^2 + 4k^2 \)
- \( 25k^2 = 13k^2 \)
4. This is not true, so this ratio does not represent a right-angled triangle.

### Conclusion
The only ratio that can represent the sides of a right-angled triangle is **Option B: 13:12:5**.

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