The inradius of the triangle having sides 26,28,30 units is

Answer

To find the inradius of a triangle with sides 26, 28, and 30 units, we can follow these steps:

### Step 1: Calculate the Semi-Perimeter
The semi-perimeter \( s \) of a triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.

For our triangle:
- \( a = 26 \)
- \( b = 28 \)
- \( c = 30 \)

Calculating the semi-perimeter:
\[ s = \frac{26 + 28 + 30}{2} = \frac{84}{2} = 42 \text{ units} \]

### Step 2: Calculate the Area of the Triangle
The area \( A \) of a triangle can be calculated using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values:
\[ A = \sqrt{42(42 - 26)(42 - 28)(42 - 30)} \]
Calculating each term:
- \( s - a = 42 - 26 = 16 \)
- \( s - b = 42 - 28 = 14 \)
- \( s - c = 42 - 30 = 12 \)

Now substituting these values into the area formula:
\[ A = \sqrt{42 \times 16 \times 14 \times 12} \]

### Step 3: Simplify the Area Calculation
We can simplify the multiplication:
\[ A = \sqrt{42 \times 16 \times 14 \times 12} \]

Factoring the terms:
- \( 42 = 6 \times 7 \)
- \( 16 = 4 \times 4 \)
- \( 14 = 7 \times 2 \)
- \( 12 = 2 \times 6 \)

Combining these factors:
\[ A = \sqrt{(6 \times 7) \times (4 \times 4) \times (7 \times 2) \times (2 \times 6)} \]
\[ = \sqrt{6^2 \times 7^2 \times 4^2 \times 2^2} \]
\[ = 6 \times 7 \times 4 \times 2 = 336 \text{ square units} \]

### Step 4: Calculate the Inradius
The inradius \( r \) can be calculated using the formula:
\[ r = \frac{A}{s} \]
Substituting the values we found:
\[ r = \frac{336}{42} = 8 \text{ units} \]

### Final Answer
The inradius of the triangle is \( 8 \) units.

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