Find whether the sides of the triangle,as given below form a right triangle or not
(1) 9 cm, 12 cm and 15 cm

Answer

To determine whether the sides of the triangle with lengths 9 cm, 12 cm, and 15 cm form 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 given sides are 9 cm, 12 cm, and 15 cm.
- Among these, the longest side is 15 cm.

2. **Assign the sides**:
- Let \( a = 9 \) cm (one side)
- Let \( b = 12 \) cm (another side)
- Let \( c = 15 \) cm (hypotenuse)

3. **Apply the Pythagorean theorem**:
- According to the theorem, for a right triangle, the following equation must hold:
\[ c^2 = a^2 + b^2 \]

4. **Calculate \( c^2 \)**:
- Calculate \( 15^2 \):
\[ c^2 = 15^2 = 225 \]

5. **Calculate \( a^2 + b^2 \)**:
- Calculate \( 9^2 \) and \( 12^2 \):
\[ a^2 = 9^2 = 81 \]
\[ b^2 = 12^2 = 144 \]
- Now add these two results:
\[ a^2 + b^2 = 81 + 144 = 225 \]

6. **Compare the results**:
- Now we compare \( c^2 \) and \( a^2 + b^2 \):
\[ 225 = 225 \]

7. **Conclusion**:
- Since \( c^2 = a^2 + b^2 \), the sides 9 cm, 12 cm, and 15 cm do indeed form a right triangle.

### Final Answer:
The triangle with sides 9 cm, 12 cm, and 15 cm is a right triangle.