`Delta ABC` is right-angled at `C.` If `AC = 5 cm and BC = 12 cm` find the length of `AB.`

Options

A. 17
B. 7
C. 13
D. 14

Explanation

To find the length of side \( AB \) in triangle \( ABC \) which is right-angled at \( C \), we will use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) 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:**
- Given: \( AC = 5 \, \text{cm} \) (one leg)
- Given: \( BC = 12 \, \text{cm} \) (the other leg)
- We need to find: \( AB \) (the hypotenuse)

2. **Apply the Pythagorean theorem:**
\[ AB^2 = AC^2 + BC^2 \]

3. **Substitute the known values into the equation:**
\[ AB^2 = (5 \, \text{cm})^2 + (12 \, \text{cm})^2 \]

4. **Calculate the squares of the lengths:**
\[ AB^2 = 25 \, \text{cm}^2 + 144 \, \text{cm}^2 \]

5. **Add the squares together:**
\[ AB^2 = 169 \, \text{cm}^2 \]

6. **Take the square root of both sides to find \( AB \):**
\[ AB = \sqrt{169 \, \text{cm}^2} \]

7. **Calculate the square root:**
\[ AB = 13 \, \text{cm} \]

### Final Answer:
The length of \( AB \) is \( 13 \, \text{cm} \).

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