Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
(i) 13 cm, 12 cm, 5 cm (ii) 20 cm, 25 cm, 30 cm.

Answer

To determine which of the given triangles are right triangles, we will use the Pythagorean theorem. According to this theorem, 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 other two sides.

### Step-by-Step Solution:

**(i) For the triangle with sides 13 cm, 12 cm, and 5 cm:**

1. **Identify the longest side:**
- The longest side is 13 cm. We will consider this as the hypotenuse (c).
- The other two sides are 12 cm (a) and 5 cm (b).

2. **Apply the Pythagorean theorem:**
- According to the theorem:
\[ a^2 + b^2 = c^2 \]
- Substitute the values:
\[ 12^2 + 5^2 = 13^2 \]
- Calculate:
\[ 144 + 25 = 169 \]
\[ 169 = 169 \]

3. **Conclusion:**
- Since both sides of the equation are equal, the triangle with sides 13 cm, 12 cm, and 5 cm is a right triangle.
- The length of the hypotenuse is **13 cm**.

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**(ii) For the triangle with sides 20 cm, 25 cm, and 30 cm:**

1. **Identify the longest side:**
- The longest side is 30 cm. We will consider this as the hypotenuse (c).
- The other two sides are 20 cm (a) and 25 cm (b).

2. **Apply the Pythagorean theorem:**
- According to the theorem:
\[ a^2 + b^2 = c^2 \]
- Substitute the values:
\[ 20^2 + 25^2 = 30^2 \]
- Calculate:
\[ 400 + 625 = 900 \]
\[ 1025 \neq 900 \]

3. **Conclusion:**
- Since both sides of the equation are not equal, the triangle with sides 20 cm, 25 cm, and 30 cm is **not a right triangle**.

### Final Results:
- The triangle with sides **13 cm, 12 cm, and 5 cm** is a right triangle with a hypotenuse of **13 cm**.
- The triangle with sides **20 cm, 25 cm, and 30 cm** is **not a right triangle**.

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