The three sides of a triangles are given which one of the following is not a right angle :

Options

A. 20, 21, 29
B. 16, 63, 65
C. 56, 90, 106
D. 36,35, 74

Answer

To determine which set of triangle sides does not form a right-angled triangle, we will use the Pythagorean theorem. The theorem states that in a right 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.

### Step-by-Step Solution:

1. **Identify the sides of the triangles:**
- Option 1: 20, 21, 29
- Option 2: 16, 63, 65
- Option 3: 56, 90, 106
- Option 4: 36, 35, 74

2. **Check Option 1: 20, 21, 29**
- Identify the longest side: 29 (hypotenuse).
- Apply the Pythagorean theorem:
\[ 29^2 = 20^2 + 21^2 \]
\[ 841 = 400 + 441 \]
\[ 841 = 841 \quad \text{(True)} \]
- Conclusion: This is a right-angled triangle.

3. **Check Option 2: 16, 63, 65**
- Identify the longest side: 65 (hypotenuse).
- Apply the Pythagorean theorem:
\[ 65^2 = 16^2 + 63^2 \]
\[ 4225 = 256 + 3969 \]
\[ 4225 = 4225 \quad \text{(True)} \]
- Conclusion: This is a right-angled triangle.

4. **Check Option 3: 56, 90, 106**
- Identify the longest side: 106 (hypotenuse).
- Apply the Pythagorean theorem:
\[ 106^2 = 56^2 + 90^2 \]
\[ 11236 = 3136 + 8100 \]
\[ 11236 = 11236 \quad \text{(True)} \]
- Conclusion: This is a right-angled triangle.

5. **Check Option 4: 36, 35, 74**
- Identify the longest side: 74 (hypotenuse).
- Apply the Pythagorean theorem:
\[ 74^2 = 36^2 + 35^2 \]
\[ 5476 = 1296 + 1225 \]
\[ 5476 = 2521 \quad \text{(False)} \]
- Conclusion: This is **not** a right-angled triangle.

### Final Answer:
The option that does not form a right-angled triangle is **Option 4: 36, 35, 74**.

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