`DeltaABC` is an isosceles triangle with `AB=AC=13cm`. AD is the median on BC from A such that `AD=12cm`. The length of BC is equal to :

Options

A. `5cm`
B. `7.5cm`
C. `10cm`
D. `6cm`

Answer

To find the length of \( BC \) in the isosceles triangle \( \Delta ABC \) where \( AB = AC = 13 \, \text{cm} \) and \( AD = 12 \, \text{cm} \) (with \( D \) being the midpoint of \( BC \)), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values:**
- We have an isosceles triangle \( \Delta ABC \) with \( AB = AC = 13 \, \text{cm} \).
- The median \( AD \) from vertex \( A \) to side \( BC \) is given as \( 12 \, \text{cm} \).

2. **Set Up the Problem:**
- Let \( BC = 2x \). Since \( D \) is the midpoint of \( BC \), we have \( BD = CD = x \).

3. **Apply the Pythagorean Theorem:**
- In triangle \( \Delta ADB \), we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
- Substituting the known values:
\[ 13^2 = 12^2 + x^2 \]

4. **Calculate the Squares:**
- Calculate \( 13^2 \) and \( 12^2 \):
\[ 169 = 144 + x^2 \]

5. **Solve for \( x^2 \):**
- Rearranging the equation gives:
\[ x^2 = 169 - 144 \]
\[ x^2 = 25 \]

6. **Find \( x \):**
- Taking the square root of both sides:
\[ x = 5 \]

7. **Calculate the Length of \( BC \):**
- Since \( BC = 2x \):
\[ BC = 2 \times 5 = 10 \, \text{cm} \]

### Conclusion:
The length of \( BC \) is \( 10 \, \text{cm} \).