Which of the following triplets will not form a right angled triangle?

Options

A. `50,30,40`
B. `15,20,25`
C. `20,29,21`
D. `12,16,11`

Answer

To determine which triplet does not form a right-angled triangle, we will use the Pythagorean theorem. According to this theorem, for three sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the following condition must hold true:

\[ c^2 = a^2 + b^2 \]

We will check each triplet to see if it satisfies this condition.

### Step-by-Step Solution:

1. **Identify the Triplets**: We have four sets of triplets to check:
- A: (50, 30, 40)
- B: (15, 20, 25)
- C: (20, 29, 21)
- D: (12, 16, 11)

2. **Check Triplet A (50, 30, 40)**:
- Identify the largest number: \(c = 50\)
- Calculate \(c^2\):
\[ 50^2 = 2500 \]
- Calculate \(a^2 + b^2\):
\[ 30^2 + 40^2 = 900 + 1600 = 2500 \]
- Since \(2500 = 2500\), triplet A is a Pythagorean triplet.

3. **Check Triplet B (15, 20, 25)**:
- Identify the largest number: \(c = 25\)
- Calculate \(c^2\):
\[ 25^2 = 625 \]
- Calculate \(a^2 + b^2\):
\[ 15^2 + 20^2 = 225 + 400 = 625 \]
- Since \(625 = 625\), triplet B is a Pythagorean triplet.

4. **Check Triplet C (20, 29, 21)**:
- Identify the largest number: \(c = 29\)
- Calculate \(c^2\):
\[ 29^2 = 841 \]
- Calculate \(a^2 + b^2\):
\[ 20^2 + 21^2 = 400 + 441 = 841 \]
- Since \(841 = 841\), triplet C is a Pythagorean triplet.

5. **Check Triplet D (12, 16, 11)**:
- Identify the largest number: \(c = 16\)
- Calculate \(c^2\):
\[ 16^2 = 256 \]
- Calculate \(a^2 + b^2\):
\[ 12^2 + 11^2 = 144 + 121 = 265 \]
- Since \(256 \neq 265\), triplet D is **not** a Pythagorean triplet.

### Conclusion:
The triplet that does not form a right-angled triangle is **D: (12, 16, 11)**.

---