Can the lengths 30 cm, 32 cm, and 16 cm be the dimensions of a right-angled triangle ?

Answer

To determine if the lengths 30 cm, 32 cm, and 16 cm can form a right-angled triangle, we will use the Pythagorean theorem. The theorem states that 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 other two sides.

### Step-by-Step Solution:

1. **Identify the lengths of the sides**:
- Given lengths are 30 cm, 32 cm, and 16 cm.
- The longest side is 32 cm, which we will consider as the hypotenuse (C).

2. **Assign the sides**:
- Let A = 30 cm (one side)
- Let B = 16 cm (another side)
- Let C = 32 cm (hypotenuse)

3. **Apply the Pythagorean theorem**:
- According to the theorem:
\[ C^2 = A^2 + B^2 \]

4. **Calculate the left-hand side (LHS)**:
- Calculate \( C^2 \):
\[ C^2 = 32^2 = 32 \times 32 = 1024 \]

5. **Calculate the right-hand side (RHS)**:
- Calculate \( A^2 + B^2 \):
\[ A^2 = 30^2 = 900 \]
\[ B^2 = 16^2 = 256 \]
\[ A^2 + B^2 = 900 + 256 = 1156 \]

6. **Compare LHS and RHS**:
- LHS = 1024
- RHS = 1156
- Since LHS (1024) is not equal to RHS (1156), the sides do not satisfy the Pythagorean theorem.

7. **Conclusion**:
- Therefore, the lengths 30 cm, 32 cm, and 16 cm cannot form a right-angled triangle.