In a ` A B C ,` if `A B=5c m ,B C=13a n dC A=12 c m` , then the distance of vertex `' A '` from the side `B C` in (in cm) `a.(25)/(13)` b. `(60)/(13)` c. `(65)/(12)` d. `(144)/(13)`

Answer

To find the distance of vertex 'A' from the side 'BC' in triangle ABC, we can use the area of the triangle calculated in two different ways and set them equal to each other. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Given:
- \( AB = 5 \, \text{cm} \)
- \( BC = 13 \, \text{cm} \)
- \( CA = 12 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (S)
The semi-perimeter \( S \) is calculated using the formula:
\[ S = \frac{AB + BC + CA}{2} \]
Substituting the values:
\[ S = \frac{5 + 13 + 12}{2} = \frac{30}{2} = 15 \, \text{cm} \]

### Step 3: Calculate the area of triangle ABC using Heron's formula
Using Heron's formula, the area \( A \) is given by:
\[ A = \sqrt{S \cdot (S - AB) \cdot (S - BC) \cdot (S - CA)} \]
Substituting the values:
\[ A = \sqrt{15 \cdot (15 - 5) \cdot (15 - 13) \cdot (15 - 12)} \]
\[ = \sqrt{15 \cdot 10 \cdot 2 \cdot 3} \]
Calculating further:
\[ = \sqrt{15 \cdot 60} = \sqrt{900} = 30 \, \text{cm}^2 \]

### Step 4: Calculate the area of triangle ABC using base and height
The area can also be calculated using the base \( BC \) and height \( h \) (the distance from vertex A to side BC):
\[ A = \frac{1}{2} \cdot BC \cdot h \]
Substituting the values:
\[ 30 = \frac{1}{2} \cdot 13 \cdot h \]

### Step 5: Solve for height (h)
Rearranging the equation to solve for \( h \):
\[ 30 = \frac{13h}{2} \]
Multiplying both sides by 2:
\[ 60 = 13h \]
Now, dividing both sides by 13:
\[ h = \frac{60}{13} \, \text{cm} \]

### Conclusion
The distance of vertex 'A' from the side 'BC' is:
\[ \boxed{\frac{60}{13} \, \text{cm}} \]