In a `DeltaABC, b = 12` units, c = 5 units and `Delta = 30`sq. units. If d is the distance between vertex A and incentre of the triangle then the value of `d^(2)` is _____

Explanation

To solve the problem step by step, we will calculate the distance \( d \) between vertex \( A \) and the incenter of triangle \( ABC \) using the given values.

### Step 1: Identify the given values
- \( b = 12 \) units
- \( c = 5 \) units
- Area \( \Delta = 30 \) square units

### Step 2: Use the formula for the area of a triangle
The area of triangle \( ABC \) can also be expressed as:
\[ \Delta = \frac{1}{2} \times b \times c \times \sin A \]
Substituting the known values:
\[ 30 = \frac{1}{2} \times 12 \times 5 \times \sin A \]
This simplifies to:
\[ 30 = 30 \sin A \]
Thus, we find:
\[ \sin A = 1 \]

### Step 3: Determine angle \( A \)
Since \( \sin A = 1 \), we have:
\[ A = 90^\circ \quad \text{or} \quad A = \frac{\pi}{2} \text{ radians} \]

### Step 4: Use the Pythagorean theorem to find side \( a \)
In a right triangle, we can use the Pythagorean theorem:
\[ a^2 = b^2 + c^2 \]
Substituting the values:
\[ a^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Thus,
\[ a = \sqrt{169} = 13 \text{ units} \]

### Step 5: Calculate the semi-perimeter \( s \)
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{13 + 12 + 5}{2} = \frac{30}{2} = 15 \text{ units} \]

### Step 6: Calculate the distance \( d \) from vertex \( A \) to the incenter
The distance \( d \) (denoted as \( AI \)) from vertex \( A \) to the incenter \( I \) can be calculated using the formula:
\[ d = AI = \frac{\Delta}{s \cos \frac{A}{2}} \]
First, we need to find \( \cos \frac{A}{2} \):
\[ A = 90^\circ \Rightarrow \frac{A}{2} = 45^\circ \Rightarrow \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]
Now substituting the values into the distance formula:
\[ d = \frac{30}{15 \cdot \frac{\sqrt{2}}{2}} = \frac{30}{\frac{15\sqrt{2}}{2}} = \frac{30 \cdot 2}{15\sqrt{2}} = \frac{60}{15\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 7: Calculate \( d^2 \)
Now, we calculate \( d^2 \):
\[ d^2 = (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

### Final Answer
Thus, the value of \( d^2 \) is:
\[ \boxed{8} \]