In a triangle `ABC, AB=17 cm, AC=9cm, AD` is perpendicular on BC and AD=3 cm. Find the circum radius of this triangle.

Answer

To find the circumradius \( R \) of triangle \( ABC \) given \( AB = 17 \, \text{cm} \), \( AC = 9 \, \text{cm} \), and the height \( AD = 3 \, \text{cm} \) (where \( AD \) is perpendicular to \( BC \)), we can follow these steps:

### Step 1: Identify the sides of the triangle
Let:
- \( a = BC \) (the base)
- \( b = AC = 9 \, \text{cm} \)
- \( c = AB = 17 \, \text{cm} \)

### Step 2: Calculate the area of triangle \( ABC \)
The area \( \Delta \) of triangle \( ABC \) can be calculated using the formula:
\[ \Delta = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can consider \( BC \) as the base and \( AD \) as the height:
\[ \Delta = \frac{1}{2} \times a \times AD \]
Since we do not know \( a \) yet, we will express the area in terms of \( a \):
\[ \Delta = \frac{1}{2} \times a \times 3 = \frac{3a}{2} \]

### Step 3: Use the circumradius formula
The circumradius \( R \) of triangle \( ABC \) can be calculated using the formula:
\[ R = \frac{abc}{4\Delta} \]
Substituting the area \( \Delta \):
\[ R = \frac{abc}{4 \times \frac{3a}{2}} = \frac{abc}{6a} = \frac{bc}{6} \]

### Step 4: Substitute the known values
Now we substitute \( b = 9 \, \text{cm} \) and \( c = 17 \, \text{cm} \):
\[ R = \frac{9 \times 17}{6} \]

### Step 5: Calculate \( R \)
Calculating the numerator:
\[ 9 \times 17 = 153 \]
Now divide by 6:
\[ R = \frac{153}{6} = 25.5 \, \text{cm} \]

### Final Answer
Thus, the circumradius \( R \) of triangle \( ABC \) is:
\[ \boxed{25.5 \, \text{cm}} \]