ABC is a right angled triangle, right at A, A circle is inscribed in it. The lengths of two sides containing the right angle are 48 cm and 14 cm . The radius of the inscribed circle is :
A)4 cm
B)8 cm
C)6 cm
D)5 cm

Options

A. 4 cm
B. 8 cm
C. 6 cm
D. 5 cm

Answer

To find the radius of the inscribed circle (inradius) of a right-angled triangle, we can use the formula:

\[ r = \frac{a + b - c}{2} \]

where:
- \(a\) and \(b\) are the lengths of the two sides containing the right angle,
- \(c\) is the length of the hypotenuse.

### Step 1: Identify the sides of the triangle
Given:
- \(a = 48 \, \text{cm}\) (one side)
- \(b = 14 \, \text{cm}\) (the other side)

### Step 2: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} \]
Substituting the values:
\[ c = \sqrt{48^2 + 14^2} = \sqrt{2304 + 196} = \sqrt{2500} = 50 \, \text{cm} \]

### Step 3: Substitute the values into the inradius formula
Now we can substitute \(a\), \(b\), and \(c\) into the inradius formula:
\[ r = \frac{48 + 14 - 50}{2} \]

### Step 4: Simplify the expression
Calculating the numerator:
\[ 48 + 14 - 50 = 12 \]
Now divide by 2:
\[ r = \frac{12}{2} = 6 \, \text{cm} \]

### Conclusion
The radius of the inscribed circle is \(6 \, \text{cm}\).

### Final Answer
C) 6 cm

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