Is the triangle with sides 12 cm, 16 cm and 18 cm a right triangle?

Answer

To determine if a triangle with sides 12 cm, 16 cm, and 18 cm is a right triangle, we can 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 triangle**:
- The sides given are 12 cm, 16 cm, and 18 cm.
- The longest side is 18 cm, which we will consider as the hypotenuse (c).

2. **Label the sides**:
- Let \( a = 12 \) cm (one leg),
- Let \( b = 16 \) cm (the other leg),
- Let \( c = 18 \) cm (the hypotenuse).

3. **Apply the Pythagorean theorem**:
- According to the theorem:
\[ c^2 = a^2 + b^2 \]

4. **Calculate \( c^2 \)**:
- \( c^2 = 18^2 = 324 \).

5. **Calculate \( a^2 \) and \( b^2 \)**:
- \( a^2 = 12^2 = 144 \),
- \( b^2 = 16^2 = 256 \).

6. **Add \( a^2 \) and \( b^2 \)**:
- \( a^2 + b^2 = 144 + 256 = 400 \).

7. **Compare \( c^2 \) with \( a^2 + b^2 \)**:
- We have \( c^2 = 324 \) and \( a^2 + b^2 = 400 \).
- Since \( 324 \neq 400 \), the condition of the Pythagorean theorem is not satisfied.

8. **Conclusion**:
- Therefore, the triangle with sides 12 cm, 16 cm, and 18 cm is **not a right triangle**.