Sides of triangle are given below .Determine which of them are right triangles .In case of a right triangle , write the length of its hypotenuse .
13 cm , 12 cm , 5 cm

Answer

To determine if the triangle with sides 13 cm, 12 cm, and 5 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 given sides are 13 cm, 12 cm, and 5 cm. Here, the longest side is 13 cm.

2. **Assign the sides**: Let’s denote:
- \( a = 12 \) cm (one leg)
- \( b = 5 \) cm (the other leg)
- \( c = 13 \) cm (the hypotenuse)

3. **Apply the Pythagorean theorem**: According to the theorem:
\[ c^2 = a^2 + b^2 \]
Substituting the values:
\[ 13^2 = 12^2 + 5^2 \]

4. **Calculate the squares**:
- \( 13^2 = 169 \)
- \( 12^2 = 144 \)
- \( 5^2 = 25 \)

5. **Add the squares of the legs**:
\[ 12^2 + 5^2 = 144 + 25 = 169 \]

6. **Compare the results**:
Since \( 169 = 169 \), the equation holds true.

7. **Conclusion**: The triangle with sides 13 cm, 12 cm, and 5 cm is indeed a right triangle, and the length of the hypotenuse is 13 cm.

### Final Answer:
The hypotenuse of the right triangle is **13 cm**.